\betaegin{document}

\title{The equivalence of lattice and Heegaard Floer homology}

\betaegin{abstract}
 We prove N\'{e}methi's conjecture: if $Y$ is a 3-manifold which is the boundary of a plumbing of a tree of disk bundles over $S^2$, then the lattice homology of $Y$ coincides with the Heegaard Floer homology of $Y$. We also give a conjectural description of the $H_1(Y)/\mathbb{T}ors$ action when $b_1(Y)>0$.
\end{abstract}

\section{Introduction}

Heegaard Floer homology is a powerful invariant of 3-manifolds, introduced by Ozsv\'{a}th and Szab\'{o} \cite{OSDisks} \cite{OSProperties}. To a closed 3-manifold $Y$, equipped with a $\Spin^c$ structure $\mathfrak{s}\in \Spin^c(Y)$, Ozsv\'{a}th and Szab\'{o} constructed an $\mathbb{F}[U]$-module denoted $\mathit{HF}^-(Y,\mathfrak{s})$.
One defines
\[
\mathit{HF}^-(Y)=\betaigoplus_{\mathfrak{s} \in \Spin^c(Y)} \mathit{HF}^-(Y,\mathfrak{s}).
\]

An important class of 3-manifolds are the \emph{plumbed} 3-manifolds. These 3-manifolds are the boundaries of 4-manifolds obtained by plumbing disk bundles over surfaces together. In this paper, we consider plumbings of disk bundles over $S^2$, such that the plumbing is encoded by a tree.  If $G$ is a tree with integer weights at the vertices, we write $Y(G)$ for the corresponding plumbed 3-manifold. 

The manifold $Y(G)$ has a convenient surgery description as integral surgery on a link $L_G\subseteq S^3$. The link $L_G$ has one unknotted component for each vertex of $G$, and a clasp for each edge of $G$. The link $L_G$ is an iterated connected sum of Hopf links. The weights give an integral framing $\Lambda$ on $L_G$, and $Y(G)\cong S^3_{\Lambda}(L_G)$.

Much effort has gone into computing the Heegaard Floer complexes of plumbed 3-manifolds. Ozsv\'{a}th and Szab\'{o} \cite{OSPlumbed} computed the Heegaard Floer homology of 3-manifolds obtained by plumbing along a tree $G$ when $\Lambda$ is negative definite and the tree has at most one \emph{bad vertex} (which is a vertex such that the weight exceeds minus the valence). This family includes the Seifert fibered spaces. 

The computation of Ozsv\'{a}th and Szab\'{o} was formalized by N\'{e}methi \cite{NemethiAR} \cite{NemethiLattice} using inspiration from algebraic geometry. N\'{e}methi defined an $\mathbb{F}[U]$-module $\mathbb{H}\mathbb{F}(Y(G))$ called \emph{lattice homology}. Lattice homology is the homology of a combinatorially defined chain complex $\mathbb{C} \mathbb{F}(G)$. N\'{e}methi proved that for the family of \emph{almost rational} graphs, $\mathit{HF}^-(Y(G))$ and $\mathbb{H} \mathbb{F}(Y(G))$ are isomorphic. Ozsv\'{a}th, Stipsicz and Szab\'{o} generalized this to a larger family called  \emph{type-2} graphs, and proved that for general $G$ there is a spectral sequence from lattice homology to Heegaard Floer homology (almost rational graphs are \emph{type-1} graphs in Ozsv\'{a}th--Stipsicz--Szab\'{o}'s terminology). In later work, N\'{e}methi proved that if $G$ is negative definite, then $Y(G)$ is an L-space if and only if $G$ is a rational graph \cite{NemethiRational}. 

 N\'{e}methi \cite{NemethiLattice} conjectured that lattice homology and Heegaard Floer homology coincide for all negative definite plumbing trees $G$. The conjecture is also open for more general plumbing graphs. The aforementioned works \cite{OSPlumbed} \cite{NemethiAR} \cite{OSSLattice} \cite{NemethiRational} verify the conjecture when $G$ is a type-2 graph and for negative definite plumbings such that $Y(G)$ is an L-space.

We recall that $\ve{\mathit{HF}}^-(Y(G))$ denotes the module obtained by completing $\mathit{HF}^-(Y(G))$ with respect to the $U$ action. If $Y(G)$ is a rational homology 3-sphere, no information is lost by taking completions. The same holds if we restrict to torsion $\Spin^c$ structures on $Y(G)$.

In this paper, we prove the conjecture in full generality:

\betaegin{thm}\label{thm:main}
If $G$ is a plumbing tree, then there is an isomorphism of $\mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}] $-modules
 \[
 \mathbb{H}\mathbb{F}(G)\cong \ve{\mathit{HF}}^-(Y(G)).
 \]
 When $b_1(Y(G))=0$, the isomorphism is relatively graded.
\end{thm}

This paper builds off previous work of the author \cite{ZemBordered} which develops a bordered theory using the Manolescu-Ozsv\'{a}th link surgery formula \cite{MOIntegerSurgery}. We note that 3-manifolds obtained by plumbing a tree of 2-spheres can be described also by gluing solid tori and cartesion products of $S^1$ and the pair-of-pants surface. In this manner we reduce the proof to local computations allowing cut and paste arguments.

\subsection{$b_1>0$}

We expect the techniques of this paper to also prove that when $b_1(Y(G))>0$ the isomorphism is relatively graded for all torsion $\Spin^c$ structures. The remaining task is to write down a theory of group valued gradings in the spirit of \cite{LOTBordered}*{Section~2.5} for the bordered link surgery modules from \cite{ZemBordered}. We plan to complete this in a future work.

When $b_1(Y)>0$, Heegaard Floer homology also has an action of $\Lambda^* H_1(Y)/\mathbb{T}ors$. In this paper, we describe a refinement of N\'{e}methi's conjecture when $b_1(Y(G))>0$. If $\gamma\in H_1(Y(G))/\mathbb{T}ors$, we define an endomorphism $\mathfrak{A}_{[\gamma]}$ on the lattice complex, and prove that our formula gives a well-defined action of $H_1(Y(G))/\mathbb{T}ors$.  See Section~\mathrm{re}f{sec:plumbed-H1-action}. We give a parallel construction on the link surgery formula in Section~\mathrm{re}f{sec:algebraic-action-link-surgery}.
 We make the following conjecture:

\betaegin{conj}
\label{conj:intro} $\ve{\mathit{HF}}^-(Y(G))$ is isomorphic to $\mathbb{HF}(G)$ as a module over $\mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}] \otimes \Lambda^* H_1(Y(G))/\mathbb{T}ors$.
\end{conj}

It is possible to extend the techniques of this paper to compute the $H_1(Y(G))/\mathbb{T}ors$-action on the Heegaard Floer homologies of plumbed manifolds in terms of the Manolescu-Ozsv\'ath link surgery formula for $L_G$. Since the techniques of \cite{ZemBordered} give a combinatorial model of the surgery formula for $L_G$, this reduces Conjecture~\mathrm{re}f{conj:intro} to a purely algebraic question.  Nonetheless, the algebraic arguments of this paper seem insufficient to show that it coincides with the action we describe on lattice homology. 

\subsection{Acknowledgments}

The author would like to thank Antonio Alfieri, Maciej Borodzik, Kristen Hendricks, Jennifer Hom, Robert Lipshitz, Beibei Liu, Ciprian Manolescu, Peter Ozsv\'{a}th and Matt Stoffregen  for helpful conversations.

\section{Background}

\subsection{Background on Heegaard Floer homology}
\label{sec:background}

In this section, we recall some background on Heegaard Floer homology and its refinements for knots and links. 

Heegaard Floer homology is an invariant of 3-manifolds equipped with a $\Spin^c$ structure $\mathfrak{s}$, defined by Ozsv\'{a}th and Szab\'{o} \cite{OSDisks} \cite{OSProperties}. Given a pointed Heegaard diagram $(\Sigma,\ve{\alphalpha},\ve{\betaeta},w)$ for $Y$, one considers the Lagrangian tori 
 \[
 \mathbb{T}_{\alpha}=\alpha_1\times \cdots \times\alpha_g\quad \text{and} \quad \mathbb{T}_{\beta}=\beta_1\times \cdots \times\beta_g,
 \]
 inside of $\Sym^g(\Sigma)$. The chain complex $\mathbb{C}F^-(Y,\mathfrak{s})$ is freely generated over $\mathbb{F}[U]$ by intersection points $\ve{x}\in \mathbb{T}_{\alpha}\cap \mathbb{T}_{\beta}$
satisfying $\mathfrak{s}_w(\ve{x})=\mathfrak{s}$. The differential counts index 1 pseudoholomorphic disks $u$ weighted by $U^{n_w(u)}$, where $n_w(u)$ is the intersection number of the image of $u$ with $\{w\}\times \Sym^{g-1}(\Sigma)$.

We now recall the construction of knot and link Floer homology. Knot Floer homology is due to Ozsv\'{a}th and Szab\'{o} \cite{OSKnots} and independently Rasmussen \cite{RasmussenKnots}. Link Floer homology is due to Ozsv\'{a}th and Szab\'{o} \cite{OSLinks}. We focus on the description in terms of a free chain complex over a 2-variable polynomial ring $\mathbb{F}[\mathscr{U},\mathscr{V}]$. Given a doubly pointed Heegaard diagram $(\Sigma,\ve{\alphalpha},\ve{\betaeta},w,z)$ representing the pair $(Y,K)$ consisting of a knot $K$ in a rational homology 3-sphere $Y$, one defines $\mathcal{C}FK(Y,K)$ to be the free $\mathbb{F}[\mathscr{U},\mathscr{V}]$-module generated by intersection points $\mathbb{T}_{\alpha}\cap \mathbb{T}_{\beta}$. The differential counts Maslov index 1 pseudo-holomorphic disks $u$ which are weighted by $\mathscr{U}^{n_{w}(u)}\mathscr{V}^{n_{z}(u)}$. 

For an $\ell$-component link $L\subseteq Y$, we consider a $2\ell$-pointed Heegaard link diagram $(\Sigma,\ve{\alphalpha},\ve{\betaeta},\ve{w},\ve{z})$ where $|\ve{w}|=|\ve{z}|=\ell$. We may similarly define $\mathcal{C}FL(Y,L)$ to be the complex freely generated over the ring $\mathbb{F}[\mathscr{U}_1,\partialots, \mathscr{U}_{\ell},\mathscr{V}_1,\partialots, \mathscr{V}_{\ell}]$ by intersection points $\ve{x}\in \mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}$. This version of link Floer homology was considered in \cite{ZemCFLTQFT}. In the differential, a holomorphic disk $u$ is weighted by the algebra element $\mathscr{U}_1^{n_{w_1}(u)}\cdots \mathscr{U}_\ell^{n_{w_\ell}(u)} \mathscr{V}_1^{n_{z_1}(u)}\cdots \mathscr{V}_\ell^{n_{z_\ell}(u)}$. 

We recall that when $L$ is a link in a rational homology 3-sphere $Y$, there is an $\ell$-component $\mathbb{Q}^\ell$-valued Alexander grading $A=(A_1,\partialots, A_\ell)$ on $\mathcal{C}FL(Y,L)$. The most important case for our purposes is when $Y=S^3$. In this case the Alexander grading takes values in the set
\[
\mathbb{H}(L):=\prod_{i=1}^\ell (\mathit{lk}(K_i,L-K_i)/2+\mathbb{Z}).
\]
 The variable $\mathscr{U}_i$ has $A_j$-grading $-\partialelta_{i,j}$ (the Kronecker delta) and $\mathscr{V}_i$ has $A_j$-grading $\partialelta_{i,j}$.

There are additional \emph{basepoint actions} on link Floer homology which make an appearance in our paper. They appear frequently when studying knot and link Floer homology (see, e.g.  \cite{SarkarMaslov} or \cite{ZemQuasi}). For each $i\in \{1,\partialots, \ell\}$, there are endomorphisms $\betaPhi_{w_i}$ and $\betaPsi_{z_i}$ of $\mathcal{C}FL(Y,L)$, defined as follows. We pick a free $\mathbb{F}[\mathscr{U}_1,\partialots, \mathscr{U}_\ell,\mathscr{V}_{1},\partialots, \mathscr{V}_{\ell}]$-basis $\ve{x}_1,\partialots, \ve{x}_n$ of $\mathcal{C}FL(Y,L)$. We write $\partial$ as an $n\times n$ matrix with this basis. The map $\betaPhi_{w_i}$ is obtained by differentiating this matrix with respect to $\mathscr{U}_i$. The map $\betaPsi_{z_i}$ is obtained by differentiating this matrix with respect to $\mathscr{V}_i$.

\subsection{Hypercubes and hyperboxes}

In this section we recall Manolescu and Ozsv\'{a}th's notion of a hypercube of chain complexes, as well as versions in the Fukaya category. See \cite{MOIntegerSurgery}*{Section~5 and 8}.
We write $\mathbb{E}_n=\{0,1\}^n$. If $\ve{d}=(d_1,\partialots, d_n)$, we write $\mathbb{E}(\ve{d})=\{0,\partialots, d_1\}\times \cdots \times \{0,\partialots, d_n\}$.
We write $\varepsilon\le \varepsilon'$ if inequality holds at all coordinates. We write $\varepsilon<\varepsilon'$ if $\varepsilon\le \varepsilon'$ and $\varepsilon\neq \varepsilon'$.

\betaegin{define} A \emph{hypercube of chain complexes} consists of a collection of groups $C_\varepsilon$, ranging over all $\varepsilon\in \mathbb{E}_n$, and maps $D_{\varepsilon,\varepsilon'}\colon C_{\varepsilon}\to C_{\varepsilon'}$, ranging over all pairs $\varepsilon,\varepsilon'$ such that $\varepsilon\le \varepsilon'$. We assume furthermore that whenever $\varepsilon\le \varepsilon''$ we have
\betaegin{equation}
\sum_{\substack{\varepsilon'\in \mathbb{E}_n \\\varepsilon\le \varepsilon'\le \varepsilon''}} D_{\varepsilon',\varepsilon''}\circ D_{\varepsilon,\varepsilon'}=0.
\label{eq:hypercube-relations}
\end{equation}
\end{define}

A \emph{hyperbox of chain complexes of size $\ve{d}\in (\mathbb{Z}^{>0})^n$} is similar. We assume that we have a collection of groups $C_{\varepsilon}$ ranging over $\varepsilon\in \mathbb{E}(\ve{d})$, as well as a collection of linear maps $D_{\varepsilon,\varepsilon'}$ ranging over all $\varepsilon\le \varepsilon'$ such that $|\varepsilon'-\varepsilon|_{L^\infty}\le 1$. We assume that Equation~\eqref{eq:hypercube-relations} holds whenever $|\varepsilon''-\varepsilon|_{L^\infty}\le 1$. 

\betaegin{rem}
The reader may find it helpful to note that the categories of hypercubes and hyperboxes are equivalent to category of type-$D$ modules over certain algebras, called the \emph{cube} and \emph{box} algebras. See \cite{ZemBordered}*{Section~3.6}.
\end{rem}

For our purposes, it is also important to consider a notion of hypercubes in the Fukaya category (see \cite{MOIntegerSurgery}*{Section~8.2}): 

\betaegin{define} A \emph{hypercube of beta-attaching curves} $\mathcal{L}_{\beta}$ on $\Sigma$ consists of a collection of attaching curves $\ve{\betaeta}_{\varepsilon}$ ranging over $\varepsilon\in \mathbb{E}_n$, as well as a collection of chains (i.e. morphisms in the Fukaya category)
\[
\mathbb{T}heta_{\varepsilon,\varepsilon'}\in \ve{\mathbb{C}F}^-(\Sigma,\ve{\betaeta}_{\varepsilon},\ve{\betaeta}_{\varepsilon'})
\]
whenever $\varepsilon<\varepsilon'$, satisfying the following compatibility condition for each pair $\varepsilon<\varepsilon'$:
\[
\sum_{\varepsilon=\varepsilon_1<\partialots<\varepsilon_j=\varepsilon'} f_{\beta_{\varepsilon_1},\partialots, \beta_{\varepsilon_j}}(\mathbb{T}heta_{\varepsilon_1,\varepsilon_2},\partialots, \mathbb{T}heta_{\varepsilon_{j-1},\varepsilon_j})=0.
\]
\end{define}
We usually assume that the diagram containing all $2^n$ attaching curves is weakly admissible. For the purposes of this paper, it is also sufficient to consider only hypercubes where each pair $\ve{\betaeta}_{\varepsilon}$ and $\ve{\betaeta}_{\varepsilon'}$ are handleslide-equivalent.

A hypercube of \emph{alpha}-attaching curves $\mathcal{L}_{\alpha}=(\ve{\alphalpha}_{\varepsilon},\mathbb{T}heta_{\varepsilon,\varepsilon'})_{\varepsilon\in \mathbb{E}_{n}}$ is defined similarly, except that we have a choice of chain $\mathbb{T}heta_{\varepsilon,\varepsilon'}\in \ve{\mathbb{C}F}^-(\Sigma,\ve{\alphalpha}_{\varepsilon'},\ve{\alphalpha}_{\varepsilon})$ whenever $\varepsilon<\varepsilon'$. 

Given hypercubes of alpha and beta attaching curves $\mathcal{L}_{\alpha}$ and $\mathcal{L}_{\beta}$, we may pair them and form a complex $\ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha},\mathcal{L}_{\beta})$, which is a hypercube of chain complexes of dimension $n+m$, where $n=\partialim \mathcal{L}_\alpha$ and $m=\partialim \mathcal{L}_{\beta}$. If $(\varepsilon,\nu)\in \mathbb{E}_n\times \mathbb{E}_m$, then the underlying chain complex of the pairing is $C_{(\varepsilon,\nu)}:=\ve{\mathbb{C}F}^-(\ve{\alphalpha}_{\varepsilon}, \ve{\betaeta}_{\nu})$. The hypercube differential is defined as follows. We set $D_{(\varepsilon,\nu),(\varepsilon,\nu)}$ to be the ordinary Floer differential. If $(\varepsilon,\nu)< (\varepsilon',\nu')$, then
\betaegin{equation}
D_{(\varepsilon,\nu),(\varepsilon',\nu')}(\ve{x})=\sum_{\substack{\varepsilon=\varepsilon_1<\partialots<\varepsilon_i=\varepsilon'\\ \nu=\nu_1<\cdots<\nu_j=\nu'}} f_{\alpha_{\varepsilon_i},\partialots, \alpha_{\varepsilon_1}, \beta_{\nu_1},\partialots, \beta_{\nu_j}}(\mathbb{T}heta_{\alpha_{\varepsilon_i},\alpha_{\varepsilon_{i-1}}},\partialots,\ve{x},\partialots, \mathbb{T}heta_{\beta_{\nu_{j-1}},\beta_{\nu_j}}).
\label{eq:hypercube-differential}
\end{equation}

\subsection{Background on lattice homology}

In this section, we recall the definition of lattice homology. Lattice homology is an invariant of plumbed 3-manifolds due to N\'{e}methi \cite{NemethiAR} \cite{NemethiLattice}, which is a formalization of Ozsv\'{a}th and Szab\'{o}'s computation of the Heegaard Floer homology of some plumbed 3-manifolds \cite{OSPlumbed}.  We will use the notation of  Ozsv\'{a}th, Stipsicz and Szab\'{o} \cite{OSSLattice} because of its relation to the surgery formula of Manolescu and Ozsv\'{a}th. 
 
 If $G$ is a plumbing tree, then we write $Y(G)$ for the associated 3-manifold and $X(G)$ for the associated 4-manifold, which has boundary $Y(G)$. We recall that a $K\in H^2(X(G))$ is a \emph{characteristic vector} if
 \[
 K (\Sigma)+\Sigma^2\equiv 0\pmod 2,
 \]
 for every class $\Sigma\in H_2(X(G))$. We write $\mathbb{C}har(G)$ for the set of characteristic vectors of $X(G)$.
 
 
We now sketch the definition of $\mathbb{CF}(G)$. Write $V(G)$ for the set of vertices, and $\mathbb{P}(G)$ for the power set of $V(G)$. Generators of the complex are written $[K,E]$ where $K\in \mathbb{C}har(X(G))$ and $E\in \mathbb{P}(G)$. The lattice complex is defined as
 \[
 \mathbb{C}F(G)=\prod_{\substack{
 K\in \mathbb{C}har(G)\\
 E\in \mathbb{P}(V)}} \mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}] \otimes \langle [K,E]\rangle.
 \]
 The differential on $\mathbb{C}F(G)$ is defined via the equation
 \betaegin{equation}
 \partial[K,E]=\sum_{v\in E} U^{a_v(K,E)}\otimes [K,E-v]+\sum_{v\in E} U^{b_v(K,E)}\otimes [K+2v^*,E-v], \label{eq:lattice-differential}
 \end{equation}
 extended equivariantly over $U$. In the above equation, $a_v(K,E)$ and $b_v(K,E)$ denote certain nonnegative integers. See \cite{OSSKnotLatticeHomology}*{Section~2} for the definition in our present notation.
 

  Ozsv\'{a}th, Stipsicz and Szab\'{o} gave an alternate description of the lattice complex which is important for our purposes.  Let $(\mathcal{C}_{\varepsilon},D_{\varepsilon,\varepsilon'})$ denote the link surgery hypercube for $L_G$. We may construct another hypercube $(H_{\varepsilon}, d_{\varepsilon,\varepsilon'})$ by setting $H_{\varepsilon}=H_* (\mathcal{C}_{\varepsilon})$, and by setting $d_{\varepsilon,\varepsilon'}=(D_{\varepsilon,\varepsilon'})_*$ if $|\varepsilon'-\varepsilon|_{L^1}=1$ and $d_{\varepsilon,\varepsilon'}=0$ otherwise. Note that $\mathcal{C}_{\varepsilon}$ is naturally a module over $\mathbb{F}[\hspace{-.5mm}[ U_1,\partialots, U_{\ell}]\hspace{-.5mm}] $, where $\ell=|L_G|$, however each $U_i$ has the same action on homology, so we view $H_* (\mathcal{C}_{\varepsilon})$ as being an $\mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}] $-module, where $U$ acts by any of the $U_i$.   Ozsv\'{a}th, Stipsicz and Szab\'{o} prove the following:
 
 \betaegin{prop}[\cite{OSSLattice}*{Proposition~4.4}]
 \label{prop:OSS-lattice}
  There is an isomorphism of hypercubes of chain complexes over $\mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}] $:
  \[
  (H_\varepsilon,d_{\varepsilon,\varepsilon'})\cong \mathbb{C} \mathbb{F}(G)
  \]
  which is relatively graded on torsion $\Spin^c$ structures. In particular, the total homologies of the two sides coincide.
 \end{prop}

\subsection{Knot and link surgery formulas}

We now recall some basics of the Manolescu--Ozsv\'{a}th link surgery formula \cite{MOIntegerSurgery}, as well as the knot surgery formulas of Ozsv\'{a}th and Szab\'{o} \cite{OSIntegerSurgeries} \cite{OSRationalSurgeries}.

If $K$ is a knot in an integer homology 3-sphere $Y$, then Ozsv\'{a}th and Szab\'{o} \cite{OSIntegerSurgeries} proved a formula which relates $\mathit{HF}^-(Y_n(K))$ with the knot Floer complex of $K$. They defined two chain complexes $\mathbb{A}(Y,K)$ and $\mathbb{B}(Y,K)$ over $\mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}] $, as well as two $\mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}] $-equivariant maps $v,h_n\colon \mathbb{A}(Y,K)\to \mathbb{B}(Y,K)$ such that
\[
\ve{\mathit{HF}}^-(Y_n(K))\cong H_* \mathbb{C}one(v+h_n\colon \mathbb{A}(Y,K)\to \mathbb{B}(Y,K)).
\]
Here, $\ve{\mathit{HF}}^-$ denotes $\mathit{HF}^-$ with coefficients in $\mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}]$ and the modules $\mathbb{A}(Y,K)$ and $\mathbb{B}(Y,K)$ are suitable completions of $\mathcal{C}FK(Y,K)$ and $\mathscr{V}^{-1} \mathcal{C}FK(Y,K)$, respectively.

The map $v$ is the canonical inclusion map, while $h_n$ is defined by composing the canonical inclusion map of $\mathcal{C}FK(Y,K)$ into $\mathscr{U}^{-1} \mathcal{C}FK(Y,K)$, and then composing with a homotopy equivalence of $\mathbb{F}[U]$-chain complexes $\mathscr{U}^{-1} \mathcal{C}FK(Y,K)\simeq \mathscr{V}^{-1}\mathcal{C}FK(Y,K)$. The map $h_n$ shifts the Alexander grading by $n$.

Manolescu and Ozsv\'{a}th extended the knot surgery formula to links in $S^3$ \cite{MOIntegerSurgery}. To a link $L\subseteq S^3$ with integral framing $\Lambda$, they constructed a chain complex $\mathcal{C}_{\Lambda}(L)$ whose homology is $\ve{\mathit{HF}}^-(S^3_{\Lambda}(L))$. The chain complex $\mathcal{C}_{\Lambda}(L)$ is built from the link Floer complex of $L$. The chain complex $\mathcal{C}_{\Lambda}(L)$ is an $\ell$-dimensional hypercube of chain complexes, where $\ell=|L|$.

We usually conflate the integral framing $\Lambda$ with the symmetric framing matrix, which has $\Lambda_{i,i}$ equal to the framing of $L_i$, and $\Lambda_{i,j}=\mathit{lk}(L_i,L_j)$ if $i\neq j$.

 The underlying group of $\mathcal{C}_{\Lambda}(L)$ has a simple description in terms of link Floer homology. If $\varepsilon\in \mathbb{E}_\ell$, write $S_{\varepsilon}$ for the multiplicatively closed subset of $\mathbb{F}[\mathscr{U}_1,\partialots, \mathscr{U}_{\ell},\mathscr{V}_1,\partialots, \mathscr{V}_\ell]$ generated by $\mathscr{V}_i$ for $i$ such that $\varepsilon_i=1$. Then $\mathcal{C}_{\varepsilon}$ is a completion of  $S_\varepsilon^{-1}\cdot \mathcal{C}FL(L).$
(This is a slight reformulation of Manolescu and Ozsv\'{a}th's original description; see \cite{ZemBordered}*{Lemma~7.5}).

The hypercube differential decomposes over sublinks of $L$ which are oriented (possibly differently than $L$). If $\vec{M}$ is an oriented sublink of $L$, we write $\betaPhi^{\vec{M}}$ for the corresponding summand of the differential. If $\varepsilon< \varepsilon'$ and $\varepsilon'-\varepsilon$ is the indicator function for the components of $M$, then $\betaPhi^{\vec{M}}$ sends $\mathcal{C}_{\varepsilon}$ to $\mathcal{C}_{\varepsilon'}$. 

For each $\varepsilon$, the group $\mathcal{C}_{\varepsilon}\cong  S_{\varepsilon}^{-1}\cdot \mathcal{C}FL(L)$ descomposes over Alexander gradings $\ve{s}\in \mathbb{H}(L)$. We write $\mathcal{C}_{\varepsilon}(\ve{s})\subseteq S_{\varepsilon}^{-1}\cdot \mathcal{C}FL(L)$ for this subgroup. The group $\mathcal{C}_{\varepsilon}(\ve{s})$ is preserved by the action of $U_i=\mathscr{U}_i\mathscr{V}_i$, for $i\in \{1,\partialots, \ell\}$, as well as the internal differential from $\mathcal{C}FL(L)$, though it is not preserved by the actions of $\mathscr{U}_i$ or $\mathscr{V}_i$.

The hypercube maps $\betaPhi^{\vec{M}}$ have a predictable effect on Alexander gradings. If $\vec{M}\subseteq L$ is an oriented sublink, define $\Lambda_{L,\vec{M}}\in \mathbb{Z}^\ell$ as follows. We may canonically identify $\mathbb{Z}^\ell$ with $H_1(S^3\setminus \nu(L))$. Under this isomorphism, the generators $(0,\partialots, 1,\partialots 0)$ of $\mathbb{Z}^\ell$ are identified with meridians of link components of $L$. We define $\Lambda_{L,\vec{M}}$ to be the sum of the longitudes of link components $K_i$ in $\vec{M}$ such that the orientations from $\vec{M}$ and $L$ are opposite. With respect to this notation, 
\betaegin{equation}
\betaPhi^{\vec{M}}(\mathcal{C}_{\varepsilon}(\ve{s}))\subseteq \mathcal{C}_{\varepsilon'}(\ve{s}+\Lambda_{L,\vec{M}})\label{eq:grading-link-surgery}
\end{equation}
where $\varepsilon'-\varepsilon$ is the indicator function for $M$.

\subsection{Systems of arcs}
\label{sec:system-of-arcs}

In this section, we recall the notion of a \emph{system of arcs} for a link $L$, which is a piece of auxiliary data necessary to build the link surgery formula.

\betaegin{define} Suppose that $L\subseteq S^3$ is a link such that each component $K_i\subseteq L$ is equipped with a pair of basepoints, denoted $w_i$ and $z_i$. A \emph{system of arcs} $\mathscr{A}$ for $L\subseteq S^3$ consists of a collection of $\ell=|L|$ embedded and pairwise disjoint arcs $\lambda_1,\partialots, \lambda_\ell$ arcs, such that
 \[
\lambda_i \cap L=\partial \lambda_i \cap L=\{w_i,z_i\}.
\]
\end{define}

If $L$ is oriented, we say an arc $\lambda_i$ is \emph{beta-parallel} if it is a small push-off of the segment of $K_i$ which is oriented from $z_i$ to $w_i$. We say that $\lambda_i$ is \emph{alpha-parallel} if it is a small push-off of the segment of $K_i$ oriented from $w_i$ to $z_i$.

The construction of Manolescu and Ozsv\'{a}th focuses on arc systems where all of the arcs are alpha-parallel. Their proof also applies with little change to the case that each arc is either alpha-parallel or beta-parallel. We write $\mathcal{C}_{\Lambda}(L,\mathscr{A})$ for the link surgery complex computed with the arc system $\mathscr{A}$.

In \cite{ZemBordered}*{Section~13}, the author studied the effect of changing the arc system, and proved a general formula which computes the effect of changing the arc system. See \cite{ZemBordered}*{Corollary~13.5}. A particularly important result is the following:

\betaegin{thm}[\cite{ZemBordered}*{Theorem~13.1}] Let $L\subseteq S^3$ be a framed link, and let $\mathscr{A}$ and $\mathscr{A}'$ be arc systems which differ only on a single knot component $K_1$. Then
\[
\mathcal{C}_{\Lambda}(L,\mathscr{A})\simeq \mathcal{C}_{\Lambda}(L,\mathscr{A}').
\]
Furthermore, the homotopy equivalence is equivariant with respect to $\mathbb{F}[\hspace{-.5mm}[ U_2,\partialots, U_\ell]\hspace{-.5mm}] $.
\end{thm}

There is a refinement in terms of type-$D$ modules as well. See \cite{ZemBordered}*{Proposition~13.2}.

\subsection{Construction of the surgery hypercube}
\label{sec:basic-systems}
In this section, we sketch the construction of the link surgery hypercube. Manolescu and Ozsv\'{a}th's construction requires a large collection of Heegaard diagrams, which they refer to as a \emph{complete system of Heegaard diagrams}. We describe their construction in a restricted setting, focusing on the case that each arc in our system of arcs $\mathscr{A}$ for $L$ is either alpha-parallel or beta-parallel. 

We also focus our attention on a special 
class of complete systems, which we call \emph{meridional $\sigma$-basic systems of Heegaard diagrams}.  Such a basic system is constructed via the following procedure. We begin with a Heegaard link diagram $(\Sigma,\ve{\alphalpha},\ve{\betaeta},\ve{w},\ve{z})$ of $(S^3,L)$. If $K_i$ is a component of $L$, write $\lambda_i$ for the arc of $\mathscr{A}$ for $K_i$. We assume that this Heegaard diagram is chosen so that the basepoints $w_i,z_i$ on  $K_i$ are separated by a single alpha curve $\alpha_i^s$ (if $\lambda_i$ is beta-parallel) or a single beta curve $\beta_i^s$ (if $\lambda_i$ is alpha-parallel). Furthermore, we assume that the arc $\lambda_i$ is embedded in $\Sigma$, and $\lambda_i$ is disjoint from all of the attaching curves except for the special meridional alpha or beta curve of $K_i$.  See Figure~\mathrm{re}f{cor:2}.

Suppose $\lambda_i$ is beta-parallel. We consider the component $A_i\subseteq \Sigma\setminus \ve{\alphalpha}$ which contains $w_i$ and $z_i$. If we glue the two boundary components of $A_i$ corresponding to $\alpha_i^s$, we obtain a torus with many disks removed (corresponding to other alpha curves). We write $\ve{\alphalpha}_0\subseteq \ve{\alphalpha}$ and $\ve{\betaeta}_0\subseteq \ve{\betaeta}$ for the curves which are not special meridians of any component. See Figure~\mathrm{re}f{cor:2}.

\betaegin{figure}[ht]
\input{cor2.pdf_tex}
\caption{The region $A_i\subseteq \Sigma$ in a meridional $\sigma$-basic system. The arc $\lambda_i$ is the dashed arc which connects $z_i$ and $w_i$. The dashed curves labeled $\alpha'$ denote the successive replacements of $\alpha^s_i$ in the basic system.}
\label{cor:2}
\end{figure}

Given such a diagram $(\Sigma,\ve{\alphalpha},\ve{\betaeta},\ve{w},\ve{z})$, we may construct two generalized hyperboxes of attaching curves $\mathcal{L}_{\alpha}$ and $\mathcal{L}_{\beta}$, as follows. The hyperboxes are generalized in the sense that each subcube is allowed to have some axis directions such that all morphisms in this direction are length 1 and consist of a canonical diffeomorphism map for moving $z_i$ to $w_i$ along $\lambda_i$, instead of a Floer chain as in a normal hypercube of attaching curves. Each of the axis directions of $\mathcal{L}_{\alpha}$ and $\mathcal{L}_{\beta}$ is identified with a different component of $L$.

We form the $K_i$-direction of $\mathcal{L}_{\alpha}$ as follows. The first step corresponds to performing a surface isotopy which moves $z_i$ to $w_i$ along the arc $\lambda_i$. This changes only the special meridian $\alpha_i^s$. The subsequent steps in the $K_i$-direction correspond to moving $\alpha_i^s$ in the component $A_i$, while avoiding the basepoint $w_i$, so that it returns to its original position. This is achieved by a sequence of isotopies and handleslides of $\alpha_i^s$ across the other components of $\ve{\alphalpha}_0$. Since all of the $A_i$ have disjoint interiors, we may do this independently for each link component of $L$ which has a beta-parallel arc. Each of the curves of $\ve{\alphalpha}_0$ is stationary in this construction. To form the hyperbox $\mathcal{L}_{\alpha}$, we perform small Hamiltonian translations to each curve of $\ve{\alphalpha}_0$ to achieve admissibility.  The construction of $\mathcal{L}_{\beta}$ is similar.

By pairing $\mathcal{L}_{\alpha}$ and $\mathcal{L}_{\beta}$ together, we obtain an $\ell$-dimensional hyperbox of chain complexes, which we compress to obtain an $\ell$-dimensional hypercube of chain complexes, which we will denote by $(\mathscr{C}_{\varepsilon}, \mathscr{D}_{\varepsilon,\varepsilon'})_{\varepsilon\in \mathbb{E}_\ell}$. We may view each $\mathscr{C}_{\varepsilon}$ as being the Floer complex obtained from $(\Sigma,\ve{\alphalpha},\ve{\betaeta},\ve{w},\ve{z})$ by keeping one basepoint from each link component, as follows. If $\varepsilon\in \mathbb{E}_\ell$, define the submodule 
\[
N_{\varepsilon}\subseteq \mathcal{C}FL(L)
\]
to be generated by $(\mathscr{V}_{\varepsilon_i}-1)\cdot \mathcal{C}FL(L)$, ranging over $i$ such that $\varepsilon_i=1$, as well as $(\mathscr{U}_{\varepsilon_i}-1)\cdot \mathcal{C}FL(L)$, ranging over $i$ such that $\varepsilon_i=0$. Then 
\[
\mathscr{C}_{\varepsilon}\cong \mathcal{C}FL(L)/N_{\varepsilon}.
\]

The link surgery hypercube $\mathcal{C}_{\Lambda}(L)$ is defined as follows. If $K_i\subseteq L$ is a link component (oriented positively), then  $\betaPhi^{+K_i}$ is defined to be the canonical map for localizing at $\mathscr{V}_i$. If $\vec{M}\subseteq L$ is a sublink, all of whose components are oriented oppositely to $L$, and $\varepsilon',\varepsilon\in \mathbb{E}_\ell$ are points such that $\varepsilon'-\varepsilon$ is the indicator function for $M$, then the map $\betaPhi^{\vec{M}}$ is defined to be the unique map which reduces to $\mathscr{D}_{\varepsilon,\varepsilon'}$ after quotienting the domain and codomain of $\betaPhi^{\vec{M}}$ by $N_{\varepsilon}$ and $N_{\varepsilon'}$, respectively, and which satisfies the grading property in Equation~\eqref{eq:grading-link-surgery}. One sets $\betaPhi^{\vec{M}}=0$ if $\vec{M}$ has more than one component and $\vec{M}$ has a component oriented coherently with $L$.

 

\section{The bordered perspective on the link surgery formula}
\label{sec:bordered}
In this section, we  recall the bordered perspective \cite{ZemBordered} on the knot and link surgery formulas \cite{OSKnots} \cite{MOIntegerSurgery}. We also prove several important new properties.

\subsection{Linear topological spaces}

In this section, we recall some preliminaries about completions. These are used in the module categories from \cite{ZemBordered}*{Section~6}. We refer the reader to \cite{AtiayhMacdonald}*{Section~10} for more background.  A \emph{linear topological vector space} $\mathcal{X}$ consists of a vector space equipped with a topology such that the following hold:
\betaegin{enumerate}
\item There is a basis of open sets centered at $0$ consisting of subspaces.
\item The addition function $\mathcal{X}\times \mathcal{X}\to \mathcal{X}$ is continuous.
\end{enumerate}

Such a topology may be specified by picking a decreasing filtration $(\mathcal{X}_{\alpha})_{\alpha\in A}$ of subspaces of $\mathcal{X}$, indexed by some directed, partially ordered set $A$. Every linear topological space can be expressed this way, for example by setting $A$ to be the set of open subspaces of $\mathcal{X}$, ordered by reverse inclusion.

 If $\mathcal{X}$ has filtration $(\mathcal{X}_{\alpha})_{\alpha\in A}$, as above, then the \emph{completion} of $\mathcal{X}$ is the inverse limit 
\[
\ve{\mathcal{X}}=\varprojlim_{\alpha\in A} \mathcal{X}/\mathcal{X}_{\alpha}.
\]

If $R$ is a ring, a \emph{linear topological $R$-module} is similar, except we require a basis at $0$ to consist of $R$-submodules.

Given two linear topological vector spaces (or linear topological $R$-modules) $\mathcal{X}$ and $\mathcal{Y}$, there are several ways to topologize the tensor product. This phenomenon parallels notions from functional analysis, where the tensor product of two Banach spaces possesses many different Banach space structures. See \cite{GrothendieckNuclear}.  We consider the following topologies:
\betaegin{enumerate}
\item $\mathcal{X}\otimes^! \mathcal{Y}$ (the \emph{standard tensor product}): A subspace $E\subseteq \mathcal{X}\otimes^! \mathcal{Y}$ is open if and only if there are open subspaces $U\subseteq \mathcal{X}$ and $V\subseteq \mathcal{Y}$ so that $U\otimes \mathcal{Y}$ and $\mathcal{X}\otimes V$ are contained in $E$.
\item $\mathcal{X}\mathrel{\vec{\otimes}} \mathcal{Y}$: A subspace $E\subseteq \mathcal{X}\mathrel{\vec{\otimes}} \mathcal{Y}$ is open if and only there is some open subspace $U\subseteq \mathcal{X}$ such that $U\otimes \mathcal{Y}\subseteq E$, and for all $x\in X$ there is an open subspace $V_x\subseteq \mathcal{Y}$ so that $x\otimes V_x\subseteq E$.
\item $\mathcal{X}\otimes^*\mathcal{Y}$: A subspace $E\subseteq \mathcal{X}\otimes^*\mathcal{Y}$ is open if and only if the following hold: there are open subspaces $U\subseteq \mathcal{X}$ and $V\subseteq \mathcal{Y}$ such that $U\otimes V\subseteq E$; for each $x\in \mathcal{X}$, there is an open subspace $V_x\subseteq \mathcal{Y}$ so that $x\otimes V_x\subseteq E$; and for all $y\in \mathcal{Y}$ there is an open subspace $U_y\subseteq \mathcal{X}$ so that $U_y\otimes y\subseteq E$.
\end{enumerate}
The above are described in Beilinson \cite{Beilinson-Tensors}. See Positelski's work \cite{Positelski-Linear} for a helpful introduction. The `standard tensor product' predates Beilinson's work and coincides with well-known constructions. The second two are due to Beilinson.

\subsection{The algebra $\mathcal{K}$}
\label{sec:surgery-algebra}

  The author described in \cite{ZemBordered} the following associative algebra $\mathcal{K}$. The algebra $\mathcal{K}$ is an algebra over the ring of two idempotents $\ve{I}_0\oplus \ve{I}_1$, where $\ve{I}_{\varepsilon}\cong \mathbb{F}$ for $\varepsilon\in \{0,1\}$. We set
\[
\ve{I}_0\cdot \mathcal{K}\cdot \ve{I}_0=\mathbb{F}[\mathscr{U},\mathscr{V}]\quad\text{and} \quad \ve{I}_1\cdot \mathcal{K}\cdot \ve{I}_1=\mathbb{F}[\mathscr{U},\mathscr{V},\mathscr{V}^{-1}].
\]
Furthermore, $\ve{I}_0\cdot \mathcal{K} \cdot \ve{I}_1=0$. Finally, $\ve{I}_1\cdot \mathcal{K}\cdot \ve{I}_0$ has two special algebra elements, $\sigma$ and $\tau$, which are subject to the relations
\[
\sigma\cdot \mathscr{U}=U \mathscr{V}^{-1} \cdot \sigma, \quad \sigma \cdot \mathscr{V}=\mathscr{V}\cdot \sigma,\quad \tau\cdot \mathscr{U}=\mathscr{V}^{-1} \cdot \tau,\quad \text{and} \quad \tau \cdot \mathscr{V}=U \mathscr{V} \cdot \tau,
\]
where $U=\mathscr{U}\mathscr{V}$.

It is sometimes helpful to consider the two algebra homomorphisms $I,T\colon \mathbb{F}[\mathscr{U},\mathscr{V}]\to \mathbb{F}[\mathscr{U},\mathscr{V},\mathscr{V}^{-1}]$ where $I$ is the inclusion given by localizing at $\mathscr{V}$, and $T$ satisfies $T(\mathscr{U})=\mathscr{V}^{-1}$ and $T(\mathscr{V})=U \mathscr{V}$. Then the relations for $\mathcal{K}$ become
\[
\sigma\cdot a=I(a)\cdot \sigma\quad \text{and} \quad \tau\cdot a=T(a)\cdot \tau,
\]
whenever $a\in \ve{I}_0\cdot \mathcal{K}\cdot \ve{I}_0$. 

\subsection{Modules over $\mathcal{K}$}

As described in \cite{ZemBordered}*{Section~6.2}, the knot algbera $\mathcal{K}$ has a natural filtration consisting of the following subspaces:

\betaegin{define}
\label{def:knot-topology}
Suppose that $n\in \mathbb{N}$ is fixed. We define $J_n\subseteq \mathcal{K}$ to be the span of following set of generators:
\betaegin{enumerate}
\item In $\ve{I}_0\cdot \mathcal{K}\cdot \ve{I}_0$, the generators $\mathscr{U}^i\mathscr{V}^j$, for $i\gammae n$ or $j\gammae n$ (i.e.  $\max(i,j)\gammae n$).
\item In $\ve{I}_1\cdot \mathcal{K}\cdot \ve{I}_0$, the generators $\mathscr{U}^i\mathscr{V}^j \sigma$ for $i\gammae n$ or $j\gammae n$.
\item In $\ve{I}_1\cdot\mathcal{K}\cdot \ve{I}_0$, the generators $\mathscr{U}^i\mathscr{V}^j\tau$ for 
$j\le 2i-n$ or $i\gammae n$.
\item In $\ve{I}_1\cdot\mathcal{K}\cdot \ve{I}_1$, the generators $\mathscr{U}^i\mathscr{V}^j$ where $i\gammae n$.
\end{enumerate}
\end{define}

In \cite{ZemBordered}*{Proposition~6.4} the author proves that multiplication is continuous as a map
\[
\mu_2\colon \mathcal{K}\mathrel{\vec{\otimes}}_{\mathrm{I}} \mathcal{K}\to \mathcal{K}.
\]
\betaegin{rem} The map $\mu_2$ is not continuous as a map from $\mathcal{K}\otimes^!_{\mathrm{I}} \mathcal{K}$ to $\mathcal{K}$ (i.e. using the standard tensor product topology). To see this, observe that $\mathscr{V}^{-i}\otimes \mathscr{V}^i\sigma\to 0$ in $\mathcal{K}\otimes^!_{\mathrm{I}} \mathcal{K}$, while $\mu_2(\mathscr{V}^{-i}\otimes \mathscr{V}^i\sigma)=\sigma\not\to 0$. 
\end{rem}

If $\mathcal{X}$ and $\mathcal{Y}$ are linear topological spaces, we will define a \emph{linear topological morphism} from $\mathcal{X}$ to $\mathcal{Y}$ to be a continuous linear map from $\ve{\mathcal{X}}$ to $\ve{\mathcal{Y}}$.
\betaegin{define}
\label{def:Alexander-type-D}
A type-$D$ Alexander module over $\mathcal{K}$ consists of a linear topological $\ve{I}$-module $\mathcal{X}$ equipped with a linear topological morphism $\partialelta^1\colon \mathcal{X}\to \mathcal{X}\mathrel{\vec{\otimes}} \mathcal{K}$, such that $(\id\otimes \mu_2)\circ(\partialelta^1\otimes \id_{\mathcal{K}})\circ \partialelta^1=0$.
\end{define}

In \cite{ZemBordered}*{Section~6}, the author also describes the categories of type-$A$ and $DA$ Alexander modules. A type-$A$ Alexander module $(\mathcal{X}, m_j)$ consists of a linear topological right $\ve{I}$-module $\mathcal{X}$, equipped with linear topological morphisms
\[
m_{j+1}\colon \mathcal{K}\mathrel{\vec{\otimes}}_{\mathrm{I}}\cdots \mathrel{\vec{\otimes}}_{\mathrm{I}} \mathcal{K}\mathrel{\vec{\otimes}}_{\mathrm{I}} \mathcal{X}\to \mathcal{X}
\]
which satisfy the $A_\infty$-module relations. Type-$DA$ Alexander modules are similar. 

\subsection{Surgery formulas and $\mathcal{K}$}

The author showed how to view the knot and link surgery formulas of Ozsv\'{a}th--Szab\'{o} \cite{OSIntegerSurgeries} \cite{OSRationalSurgeries} and Manolescu--Ozsv\'{a}th \cite{MOIntegerSurgery} as certain types of $\mathcal{K}$-modules. The author used the algebraic framework of type-$A$ and type-$D$ modules of Lipshitz, Ozsv\'{a}th and Thurston \cite{LOTBordered} \cite{LOTBimodules}. To a knot $K$ in $S^3$ (or more generally, in a rational homology 3-sphere) the mapping cone formula of Ozsv\'{a}th--Szab\'{o} may naturally be viewed as either a type-$D$ module over $\mathcal{K}$, denoted $\mathcal{X}_{\lambda}(K)^{\mathcal{K}}$, or a type-$A$ module over $\mathcal{K}$, denoted ${}_{\mathcal{K}} \mathcal{X}_{\lambda}(K)$. Similarly, if $L\subseteq S^3$ is a link with framing $\Lambda$, the link surgery formula of Manolescu and Ozsv\'{a}th \cite{MOIntegerSurgery} may naturally be viewed as a type-$D$ module $\mathcal{X}_{\Lambda}(L)^{\mathcal{L}_{\ell}}$ over the tensor product algebra $\mathcal{L}_\ell=\mathcal{K}^{\otimes_{\mathbb{F}} \ell}$, where $\ell=|L|$.
 
 

We describe the correspondence for the case of the knot surgery formula for a knot $K\subseteq S^3$. We recall that Ozsv\'{a}th and Szab\'{o}'s mapping cone complex is an appropriate completion of the mapping cone
\[
\mathbb{C}one\left(v+h_n\colon \mathcal{C}FK(K)\to \mathscr{V}^{-1} \mathcal{C}FK(K)\right).
\]
Here,  $v$ and $h_n$ are two $\mathbb{F}[U]$-equivariant maps, where $U$ acts by $\mathscr{U}\mathscr{V}$.

 The type-$D$ module $\mathcal{X}_{n}(K)^{\mathcal{K}}$ is defined as follows. Let $\ve{x}_1,\partialots, \ve{x}_n$ be a free $\mathbb{F}[\mathscr{U},\mathscr{V}]$-basis of $\mathcal{C}FK(K)$. Then $\mathcal{X}_n(K)\cdot \ve{I}_0$ is generated over $\mathbb{F}$ by copies of the basis elements $\ve{x}_1^0,\partialots, \ve{x}_n^0$, and similarly $\mathcal{X}_n(K)\cdot \ve{I}_1$ is generated by elements $\ve{x}_1^1,\partialots, \ve{x}_n^1$. The structure map
\[
\partialelta^1\colon \mathcal{X}_n(K)\to \mathcal{X}_n(K)\otimes_{\mathrm{I}} \mathcal{K}
\]
has three types of summands: those arising from $\partial$, those arising from $v$, and those arising from $h_n$, as follows. Write $\partial$ for the differential on $\mathcal{C}FK(K)$. If $\partial\ve{x}_i$ has a summand of $ \ve{y}_j\cdot \mathscr{U}^n\mathscr{V}^m$, then $\partialelta^1(\ve{x}_i^\varepsilon)$ has a summand of $\ve{y}_j^\varepsilon\otimes \mathscr{U}^n \mathscr{V}^m$, for $\varepsilon\in \{0,1\}$. Each $\partialelta^1(\ve{x}^0_i)$ has a summand of the form $\ve{x}^1_i\otimes \sigma$. Finally, if $h_n(\ve{x}_i)$ has a summand of $ \ve{y}_j\cdot\mathscr{U}^n\mathscr{V}^m$, then $\partialelta^1(\ve{x}_i^0)$ has a summand of $\ve{y}_j^1\otimes \mathscr{U}^n \mathscr{V}^m \cdot\tau$. It is verified in \cite{ZemBordered}*{Lemma~8.9} that $\mathcal{X}_n(K)^{\mathcal{K}}$ is a type-$D$ module over $\mathcal{K}$.

For the description of the full link surgery complex of a link $L\subseteq S^3$ in terms of type-$D$ modules over the algebra $\mathcal{L}$, we refer the reader to \cite{ZemBordered}*{Section~8}. We note that in general, the homotopy type of the link surgery type-$D$ module depends non-trivially on the system of arcs $\mathscr{A}$. For example, the type-$D$ link surgery module of the Hopf link depends non-trivially on $\mathscr{A}$;  see \cite{ZemBordered}*{Section~16}. We write $\mathcal{X}_{\Lambda}(L,\mathscr{A})^{\mathcal{L}}$ for the type-$D$ module constructed with $\mathscr{A}$.

There is also a bimodule ${}_{\mathcal{K}} \mathcal{T}^{\mathcal{K}}$ whose effect is changing an alpha-parallel arc to a beta-parallel arc, and vice-versa. See \cite{ZemBordered}*{Section~14}. This bimodule is related to the Dehn twist diffeomorphism on knot Floer homology discovered by Sarkar \cite{SarkarMovingBasepoints} and further studied by the author \cite{ZemQuasi}.

\subsection{Type-$A$ and $D$ modules for solid tori}

We now review the type-$A$ and type-$D$ modules of integrally framed solid tori (by which we mean the complements of integrally framed unknots in $S^3$).

If $n\in \mathbb{Z}$, we define type-$A$ and type-$D$ modules ${}_{\mathcal{K}} \mathcal{D}_n$ and $\mathcal{D}_n^{\mathcal{K}}$ as follows. We begin with ${}_{\mathcal{K}} \mathcal{D}_n$. We set 
\[
\ve{I}_0\cdot \mathcal{D}_n=\mathbb{F}[\hspace{-.5mm}[ \mathscr{U},\mathscr{V}]\hspace{-.5mm}]\quad \text{and}\quad  \ve{I}_1\cdot \mathcal{D}_n=\mathbb{F}[\hspace{-.5mm}[ \mathscr{U},\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}].
\]
The action of $\ve{I}_{\varepsilon}\cdot \mathcal{K}\cdot \ve{I}_{\varepsilon}$ is ordinary polynomial multiplication. If $\ve{x}\in \ve{I}_0\cdot \mathcal{D}_n$,  we set $\sigma\cdot \ve{x}=I(\ve{x})$ and $\tau\cdot \ve{x}=\mathscr{V}^n\cdot T(\ve{x})$, where
\[
I,T\colon \mathbb{F}[\mathscr{U},\mathscr{V}]\to \mathbb{F}[\mathscr{U},\mathscr{V},\mathscr{V}^{-1}]
\]
are the following maps. The map $I$ is the inclusion from localizing at $\mathscr{V}$. The map $T$ is given by $T(\mathscr{U}^i\mathscr{V}^j)=\mathscr{U}^j \mathscr{V}^{2j-i}$.

The type-$D$ module $\mathcal{D}_n^{\mathcal{K}}$ is as follows. As a right $\ve{I}$-module, we set $\mathcal{D}_n\cong \ve{I}$, viewed as being generated by $i_0\in \ve{I}_0$ and $i_1\in \ve{I}_1$. The structure map is given by the formula
\[
\partialelta^1(i_0)=i_1\otimes (\sigma+\mathscr{V}^n \tau).
\]

\subsection{The merge and type-$A$ identity bimodules}

In \cite{ZemBordered}*{Section~8}, the author defined the \emph{merge} bimodule and the \emph{type-$A$ identity} bimodule. We recall the definition of these modules presently.

We begin with the merge module ${}_{\mathcal{K}| \mathcal{K}} M^{\mathcal{K}}$. Ignoring completions, the merge module is a $DA$-bimodule over $(\mathcal{K}\otimes_{\mathbb{F}} \mathcal{K}, \mathcal{K})$. However, the structure map $\partialelta_3^1$ of the merge module is not continuous as a map from $(\mathcal{K}\otimes_{\mathbb{F}}^! \mathcal{K})\mathrel{\vec{\otimes}} (\mathcal{K}\otimes_{\mathbb{F}}^! \mathcal{K})\mathrel{\vec{\otimes}} M$ to $M\mathrel{\vec{\otimes}} \mathcal{K}$. Instead, the merge module is a \emph{split Alexander} module in the terminology of \cite{ZemBordered}*{Section~6.4}. This is weaker than being an Alexander module. This condition means that the map $\partialelta_3^1$ is continuous for a finer topology than the topology used to define an Alexander module over $\mathcal{K}\otimes_{\mathbb{F}} \mathcal{K}$. The split Alexander property is sufficient for taking the box tensor product of ${}_{\mathcal{K}|\mathcal{K}} M^{\mathcal{K}}$ with a pair of type-$D$ modules $\mathcal{X}^{\mathcal{K}}$ and $\mathcal{Y}^\mathcal{K}$, but not for general type-$D$ modules over $\mathcal{K}\otimes_{\mathbb{F}}^! \mathcal{K}$. 

 As an $(\ve{I}\otimes \ve{I},\ve{I})$-module, $M$ is isomorphic to $\ve{I}$. The right action is the obvious one. The left action is given by $(i_{\varepsilon}\otimes i_{\nu})\cdot i=i_{\varepsilon}\cdot i_{\nu}\cdot i$.
The map $\partialelta_2^1$ is defined as follows. If $a,b\in \ve{I}_{0} \cdot \mathcal{K} \cdot \ve{I}_0$, then we set $\partialelta_2^1(a\otimes b, i_0)=i_0\otimes a\cdot b$. We use the same formula if  $a,b\in \ve{I}_1\cdot \mathcal{K} \cdot \ve{I}_1$. If $a$ and $b$ are in other idempotents, then we set $\partialelta_2^1(a\otimes b,i_\varepsilon)=0$. 

 Additionally, there is a $\partialelta_3^1$ term, determined by the equations
\[
\partialelta_3^1(\tau\otimes 1, 1\otimes \tau, i_0)=i_0\otimes\tau,\quad \quad \partialelta_3^1(\sigma\otimes 1, 1\otimes \sigma, i_0)=i_1\otimes \sigma \quad \text{and}
\]
\[
\partialelta_3^1(1\otimes \tau,\tau\otimes 1, i_0)=\partialelta_3^1(1\otimes \sigma,\sigma\otimes 1, i_0)=0.
\]

Using the merge module, we define the type-$A$ identity bimodule
\[
{}_{\mathcal{K}|\mathcal{K}} [\mathbb{I}^{\Supset}]:={}_{\mathcal{K}| \mathcal{K}} M^{\mathcal{K}}\mathrel{\hat{\betaoxtimes}} {}_{\mathcal{K}} \mathcal{D}_0.
\]
The type-$A$ identity modules relates the type-$D$ and type-$A$ modules of a knot complement $K$ via the formula
\[
\mathcal{X}_n(K)^\mathcal{K} \mathrel{\hat{\betaoxtimes}} {}_{\mathcal{K}|\mathcal{K}}  [\mathbb{I}^{\Supset}]\cong {}_{\mathcal{K}} \mathcal{X}_n(K).
\]

\subsection{A pairing theorem}

 Suppose that $L_1$ and $L_2$ are two links in $S^3$. The author gave several descriptions of the link surgery formula for $L_1\#L_2$ in terms of the link surgery formulas for $L_1$ and $L_2$. We refer to these connected sum formulas \emph{pairing theorems} since topologically performing $\lambda_1+\lambda_2$ surgery on $K_1\# K_2\subseteq S^3$ is the same as gluing the complements of $K_1$ and $K_2$ via the orientation reversing diffeomorphism which sends the meridian $\mu_1$ to $\mu_2$ and the longitude $\lambda_1$ of $K_1$ to the longitude $-\lambda_2$ of $K_2$.  See \cite{ZemBordered}*{Section~12} for more details on these pairing theorems.

We begin by describing the pairing theorem on the level of the link surgery complexes. Subsequently, we will describe the connected sum formula in terms of type-$D$ modules. Write $\mathcal{C}_{\Lambda_1}(L_1)=(\mathcal{C}_{\varepsilon}(L_1),d_{\varepsilon,\varepsilon'})$ and $\mathcal{C}_{\Lambda_2}(L_2)=(\mathcal{C}_{\nu}(L_2),\partialelta_{\nu,\nu'})$ for the link surgery formulas of links $L_1$ and $L_2$, with integral framings $\Lambda_1$ and $\Lambda_2$. Write $K_1$ and $K_2$ for the distinguished components of $L_1$ and $L_2$ (along which we take the connected sum). We assume that the link surgery complexes are computed with systems of arcs $\mathscr{A}_1$ and $\mathscr{A}_2$ so that $K_1$ has an alpha-parallel arc, and $K_2$ has a beta-parallel arc. We write $\mathscr{A}_{1\# 2}$ for the system of arcs on $L_1 \# L_2$ which coincides with $\mathscr{A}_1$ and $\mathscr{A}_2$ away from $K_1\# K_2$, and which has an arc for $K_1\# K_2$ consisting of the co-core of the connected sum band. See Figure~\mathrm{re}f{fig:4}.

\betaegin{figure}[ht]
\input{fig4.pdf_tex}
\caption{Left: arc systems $\mathscr{A}_1$ and $\mathscr{A}_2$ so that components $K_1\subseteq L_1$ and $K_2\subseteq L_2$ have arcs which are beta-parallel and alpha-parallel. Right: The arc system $\mathscr{A}_{1\#2}$ on $L_1\# L_2$.}
\label{fig:4}
\end{figure}

  
  
  
 If $\varepsilon\in \{0,1\}$, write $\mathcal{C}^{(*,\varepsilon)}(L_1)$ for the codimension one subcube of $\mathcal{C}_{\Lambda_1}(L_1,\mathscr{A}_1)$ consisting of complexes which have $K_1$-component $\varepsilon$. Define $\mathcal{C}^{(*,\varepsilon)}(L_2)$ similarly. We may view $\mathcal{C}_{\Lambda_1}(L_1,\mathscr{A}_1)$ as a mapping cone from $\mathcal{C}^{(*,0)}(L_1)$ to $\mathcal{C}^{(*,1)}(L_1)$ as follows:
 \[
 \mathcal{C}_{\Lambda_1}(L_1,\mathscr{A}_1)=\mathbb{C}one\betaig( \betaegin{tikzcd}[column sep=1.5cm] \mathcal{C}^{(*,0)}(L_1)\mathrm{a.r.}[r, "F^{K_1}+F^{-K_1}"]& \mathcal{C}^{(*,1)}(L_1)
 \end{tikzcd}\betaig).
 \]
In the above, and $F^{K_1}$ is the sum of the hypercube maps $\betaPhi^{\vec{M}}$ of $\mathcal{C}_{\Lambda_1}(L_1)$ ranging over all oriented sublinks $\vec{M}\subseteq L_1$ such that $+K_1\subseteq \vec{M}$. Similarly $F^{-K_1}$ is the sum of the hypercube maps such that $-K_1\subseteq \vec{M}$. Each $\mathcal{C}^{(*,\varepsilon)}(L_1)$ has an internal differential consisting of the sum of the hypercube maps for sublinks $\vec{M}$ which do not contain $\pm K_1$.  We may similarly view $\mathcal{C}_{\Lambda_2}(L_2)$ as a mapping cone from $\mathcal{C}^{(*,0)}(L_2)$ to $\mathcal{C}^{(*,1)}(L_2)$. 

The pairing theorem is the following:
\betaegin{thm}[\cite{ZemBordered}*{Theorem~12.1}]
\label{thm:pairing-main}
With respect to the above notation, the link surgery complex $\mathcal{C}_{\Lambda_1+\Lambda_2}(L_1\# L_2,\mathscr{A}_{1\# 2})$ is homotopy equivalent to
\[
 \betaegin{tikzcd}[column sep=4cm] \mathcal{C}^{(*,0)}(L_1)\otimes_{\mathbb{F}[\mathscr{U},\mathscr{V}]} \mathcal{C}^{(*,0)}(L_2)\mathrm{a.r.}[r, "F^{K_1}\otimes F^{K_2}+F^{-K_1}\otimes F^{-K_2}"] & \mathcal{C}^{(*,1)}(L_1)\otimes_{\mathbb{F}[\mathscr{U},\mathscr{V},\mathscr{V}^{-1}]} \mathcal{C}^{(*,1)}(L_2).
 \end{tikzcd}
\]
Here, $\Lambda_1+\Lambda_2$ is obtained by summing the framing on $K_1$ and $K_2$, and using the other framings on $L_1$ and $L_2$. Also, the differential on $\mathcal{C}^{(*,\varepsilon)}(L_1)\otimes \mathcal{C}^{(*,\varepsilon)}(L_2)$ is the ordinary differential (Leibniz rule) on the tensor product of two chain complexes. 
\end{thm}

Note that Theorem~\mathrm{re}f{thm:pairing-main} has an alternate description in terms of the type-$D$ modules. Namely, it  translates to the statement
\[
\mathcal{X}_{\Lambda_1+\Lambda_2}(L_1\#L_2,\mathscr{A}_{1\#2})^{\mathcal{L}_{\ell_1+\ell_2-1}}\simeq \left(\mathcal{X}_{\Lambda_1}(L_1,\mathscr{A}_1)^{\mathcal{L}_{\ell_1}} , \mathcal{X}_{\Lambda_2}(L_2,\mathscr{A}_2)^{\mathcal{L}_{\ell_2}}\right)\mathrel{\hat{\betaoxtimes}} {}_{\mathcal{K}| \mathcal{K}} M^{\mathcal{K}}.
\]
See \cite{ZemBordered}*{Section~12}. Note that if we ignore completions, the above box tensor product is obtained as follows: We first take the external tensor product of $\mathcal{X}_{\Lambda_1}(L_1,\mathscr{A}_1)^{\mathcal{L}_{\ell_1}}$ and $ \mathcal{X}_{\Lambda_2}(L_2,\mathscr{A}_2)^{\mathcal{L}_{\ell_2}}$, i.e. we take the tensor product of the modules over $\mathbb{F}$, and use the Leibniz rule to form the differential $\partialelta^1\otimes \id\otimes 1_{\mathcal{L}_{\ell_2}}+\id\otimes 1_{\mathcal{L}_{\ell_1}}\otimes \partialelta^1$, with tensor factors reordered. This yields a type-$D$ module over $\mathcal{L}_{\ell_1+\ell_2}$. Next, we view ${}_{\mathcal{K}|\mathcal{K}} M^{\mathcal{K}}$ as a type-$DA$ module over $(\mathcal{K}\otimes_{\mathbb{F}} \mathcal{K},\mathcal{K})$, and take the external tensor product of $M$ with the identity bimodule ${}_{\mathcal{L}_{\ell_1+\ell_2-2}}[\mathbb{I}]^{\mathcal{L}_{\ell_1+\ell_2-2}}$ to get a $DA$-bimodule over $(\mathcal{L}_{\ell_1+\ell_2},\mathcal{L}_{\ell_1+\ell_2-1})$. Finally, we compute the box tensor product as normal \cite{LOTBordered}*{Section~2.4}.

\subsection{The pair-of-pants bimodules}

The connected sum formula in Theorem~\mathrm{re}f{thm:pairing-main} requires one of the components $K_1\subseteq L_1$ and $K_2\subseteq L_2$ to have an alpha-parallel arc, and the other to have a beta-parallel arc. The output arc in $\mathscr{A}_{1\# 2}$ is neither alpha-parallel nor beta-parallel. For taking iterated connected sums of knots, we need to change the arc $\mathscr{A}_{1\# 2}$ so that the arc for $K_1\#K_2$ is either alpha or beta-parallel. A general formula for changing arcs is described in \cite{ZemBordered}*{Section~13.2}. In our present case, taking the connected sum and then changing the arc for $K_1\# K_2$ can be encoded by one of two bimodules, which are similar to the merge modules. We call these the \emph{pair-of-pants} bimodules,  and we denote them by ${}_{\mathcal{K}|\mathcal{K}} W_l^{\mathcal{K}}$ and ${}_{\mathcal{K}|\mathcal{K}} W_r^{\mathcal{K}}$.

The module ${}_{\mathcal{K}|\mathcal{K}} W_r^{\mathcal{K}}$ has $\partialelta_2^1$ and $\partialelta_3^1$ identical to the merge module. Additionally, there is a $\partialelta_5^1$, as follows
\betaegin{equation}
\partialelta_5^1(a|b, a'|b', 1|\tau,\tau|1, i_0)=i_1\otimes \mathscr{V}^{-1}\partial_{\mathscr{U}}(ab) a' \left(\mathscr{U} \partial_{\mathscr{U}}+\mathscr{V} \partial_{\mathscr{V}}\right)(b') \tau.  \label{eq:delta-5-1-merge}
\end{equation}
In the above, $\partial_{\mathscr{U}}$ and $\partial_{\mathscr{V}}$ denote the derivatives with respect to $\mathscr{U}$ and $\mathscr{V}$, respectively. The module ${}_{\mathcal{K}|\mathcal{K}}W_l^{\mathcal{K}}$ is similar, but has the role of the two tensor factors switched.

The importance of these bimodules is illustrated in the following theorem:

\betaegin{thm}[\cite{ZemBordered}*{Theorem~15.2}] Suppose that $L_1,L_2\subseteq S^3$ are two framed links with systems of arcs $\mathscr{A}_1$ and $\mathscr{A}_2$, respectively. Suppose also that we form $L_1\# L_2$ by taking the connected sum along components $K_1\subseteq L_1$ and $K_2\subseteq L_2$. Suppose that the arc for $K_1$ is alpha-parallel, and the arc for $K_2$ is beta-parallel. Let $\mathscr{A}_l$ denote the system of arcs on the connected sum which coincides with $\mathscr{A}_1$ and $\mathscr{A}_2$ away from $K_1\# K_2$ and is alpha-parallel on $K_1\#K_2$. Let $\mathscr{A}_r$ denote the system of analogous system of arcs on $L_1\#L_2$ which is instead beta-parallel on $K_1\# K_2$. Then
\[
\mathcal{X}_{\Lambda_1+\Lambda_2}(L_1\#L_2,\mathscr{A}_r)^{\mathcal{L}_{\ell_1+\ell_2-1}}\simeq (\mathcal{X}_{\Lambda_1}(L_1,\mathscr{A}_1)^{\mathcal{L}_{\ell_1}}, \mathcal{X}_{\Lambda_2}(L_2,\mathscr{A}_2)^{\mathcal{L}_{\ell_2}})\mathrel{\hat{\betaoxtimes}} {}_{\mathcal{K}|\mathcal{K}} W_{r}^{\mathcal{K}},
\] 
and similarly if $\mathscr{A}_{r}$ and $W_r$ are replaced by $\mathscr{A}_{l}$ and $W_l$.
\end{thm}

We now recall an alternate description of the pair-of-pants bimodules in terms of the link surgery complexes; see \cite{ZemBordered}*{Section~15.2}. Write $\mathcal{C}_{\Lambda_i}(L_i,\mathscr{A}_i)$ as mapping cones 
 \[
 \mathcal{C}_{\Lambda_i}(L_i,\mathscr{A}_i)=\mathbb{C}one\left( \betaegin{tikzcd}[column sep=1.5cm] \mathcal{C}^{(*,0)}(L_i)\mathrm{a.r.}[r, "F^{K_i}+F^{-K_i}"] & \mathcal{C}^{(*,1)}(L_i)
 \end{tikzcd}\right).
 \]
Then $\mathcal{C}_{\Lambda_1+\Lambda_2}(L_1\# L_2, \mathscr{A}_{l})$ may also be described as the mapping cone
\betaegin{equation}
 \betaegin{tikzcd}[column sep=8.5cm] \mathcal{C}^{(*,0)}_1\otimes \mathcal{C}^{(*,0)}_2\mathrm{a.r.}[r, "F^{K_1}\otimes F^{K_2}+\left(\id+  \mathscr{V}^{-1}(\betaPhi_{w_1}+\betaPhi_{w_2})\circ (\mathcal{A}_{[K_1]} \otimes \id)\right)\circ (F^{-K_1}\otimes F^{-K_2})"] & \mathcal{C}^{(*,1)}_1\otimes \mathcal{C}^{(*,1)}_2,
 \end{tikzcd}
 \label{eq:alternate-pairing-theorem}
\end{equation}
where $\mathcal{C}^{(*,\varepsilon)}_i$ denotes $\mathcal{C}^{(*,\varepsilon)}(L_i)$.
In the above, the maps $\betaPhi_{w_i}$ denote the analogs of the basepoint actions for the hypercubes $\mathcal{C}^{(*,1)}(L_i)$. These maps are defined similarly to basepoint actions on the ordinary link Floer complexes, except are define in the setting of hypercubes. Note that they are morphisms of hypercubes, so will generally have non-trivial components of length greater than zero (in particular, $\betaPhi_{w_i}$ is not the same as the internal basepoint action on $\mathcal{C}FL(L_i)$). The map $\mathcal{A}_{[K_1]}$ is the hypercube homology action of the curve $K_1\subseteq \Sigma_1$.  See \cite{ZemBordered}*{Sections~13.2 and 13.3} for more detail on these constructions.

The map $\mathcal{A}_{[K_1]}$ appearing in Equation~\eqref{eq:alternate-pairing-theorem} has another description which is more immediately related to the expression in Equation~\eqref{eq:delta-5-1-merge}. According to \cite{ZemBordered}*{Lemma~13.28}, there is a chain homotopy
\betaegin{equation}
\mathcal{A}_{[K_1]}\simeq \mathscr{U}\betaPhi_{w_1}+\mathscr{V} \betaPsi_{z_1}, \label{eq:homology-action-basepoint}
\end{equation}
where we view both maps as being endomorphisms of the hypercube $\mathcal{C}^{(*,1)}(L_1)$. In the above, we are writing $\betaPhi_{w_1}$ and $\betaPsi_{z_1}$ for the algebraic basepoint action of the basepoints of $K_1$ on the hypercube $\mathcal{C}^{(*,1)}(L_1)$.

\subsection{The Hopf link surgery complex}

We now recall the link surgery hypercube for the Hopf link, which was computed in \cite{ZemBordered}*{Section~16}. We recall the negative Hopf link has the following link Floer complex.
 \betaegin{equation}
\mathcal{C}FL(H)\cong \betaegin{tikzcd}[labels=description,row sep=1cm, column sep=1cm] \ve{a}& \ve{b}\mathrm{a.r.}[l, "\mathscr{U}_2"] \mathrm{a.r.}[d, "\mathscr{U}_1"]\\
\ve{c}\mathrm{a.r.}[r, "\mathscr{V}_2"] \mathrm{a.r.}[u, "\mathscr{V}_1"]& \ve{d}.
\end{tikzcd}
\label{eq:Hopf-def}
\end{equation}

 There are two models for the Hopf link surgery hypercube, depending on the choice of arc system. The models are summarized in the following proposition:
 \betaegin{prop}[\cite{ZemBordered}*{Proposition~16.1}]
 \label{prop:Hopf-link-large-model} Write $H=K_1\cup K_2$ for the negative Hopf link, and suppose that $\Lambda=(\lambda_1,\lambda_2)$ is an integral framing $H$. Let $\mathscr{A}$ be a system of arcs for $H$ where both components are alpha-parallel or both components are beta-parallel. The maps in the surgery complex $\mathcal{C}_{\Lambda}(H,\mathscr{A})$ are, up to overall homotopy equivalence, as follows:
 \betaegin{enumerate}
  \item The map $\betaPhi^{K_1}$ is the canonical inclusion of localization. The map $\betaPhi^{-K_1}$ is given by the following formula:
  \[
 \betaPhi^{-K_1} =\betaegin{tikzcd}[row sep=0cm]
 \ve{a} \mathrm{a.r.}[r,mapsto]& \ve{d}\mathscr{V}_1^{\lambda_1-1}\\
 \ve{b} \mathrm{a.r.}[r, mapsto]& 0\\
 \ve{c} \mathrm{a.r.}[r, mapsto]& \ve{b}\mathscr{V}_1^{\lambda_1+1}+\ve{c}\mathscr{V}_1^{\lambda_1}\mathscr{U}_2\\
 \ve{d} \mathrm{a.r.}[r,mapsto]& \ve{d}\mathscr{V}_1^{\lambda_1}\mathscr{U}_2.
 \end{tikzcd}
  \]
 \item The maps $\betaPhi^{K_2}$ is the canonical inclusion of localization, and $\betaPhi^{-K_2}$ is given by the following formula:
\[
\betaPhi^{-K_2}=\betaegin{tikzcd}[row sep=0cm]
\ve{a}\mathrm{a.r.}[r, mapsto] &\ve{a}\mathscr{U}_1\mathscr{V}_2^{\lambda_2}\\
 \ve{b}\mathrm{a.r.}[r, mapsto] &0\\
 \ve{c}\mathrm{a.r.}[r, mapsto] &\ve{b}\mathscr{V}_2^{\lambda_2+1}+\ve{c}\mathscr{U}_1\mathscr{V}_2^{\lambda_2} \\
 \ve{d}\mathrm{a.r.}[r, mapsto]&\ve{a}\mathscr{V}_2^{\lambda_2-1}.
 \end{tikzcd}
\]
 \item The length 2 map $\betaPhi^{-K_1\cup -K_2}$ is given by the following formula:
 \[
\betaPhi^{-K_1\cup -K_2}=\betaegin{tikzcd}[row sep=0cm]
 \ve{a}\mathrm{a.r.}[r, mapsto] &\ve{c}\mathscr{V}_1^{\lambda_1-2}\mathscr{V}_2^{\lambda_2-1}\\
 \ve{b}\mathrm{a.r.}[r, mapsto] &0\\
  \ve{c}\mathrm{a.r.}[r, mapsto] &\ve{d}\mathscr{V}_1^{\lambda_1-1} \mathscr{V}_2^{\lambda_2}\\
 \ve{d}\mathrm{a.r.}[r, mapsto]&\ve{c}\mathscr{V}_1^{\lambda_1-1} \mathscr{V}^{\lambda_2-2}_2.
 \end{tikzcd}
 \]
 The length 2 maps for other orientations of the Hopf link vanish.
 \end{enumerate}
 \end{prop}
 The above formulas are stated only for the values of the maps on the generators $\ve{a}$, $\ve{b}$, $\ve{c}$ and $\ve{d}$. Values on the rest of the complex are determined by the equivariance properties of the maps proven in \cite{ZemBordered}*{Lemma~7.7}. 

 
  \betaegin{prop}[\cite{ZemBordered}*{Proposition~16.7}] Let $\overline\mathscr{A}$ be a system of arcs on the Hopf link $H$ where one arc is alpha-parallel and the other arc is beta-parallel. The surgery complex for the Hopf link $\mathcal{C}_{\Lambda}(H,\overline\mathscr{A})$ is identical to the one in Proposition~\mathrm{re}f{prop:Hopf-link-large-model}, except that we delete the term $\ve{c}\mapsto  \mathscr{V}_1^{\lambda_1-1} \mathscr{V}_2^{\lambda_2} \ve{d}$ from the expression in $\betaPhi^{-K_1\cup -K_2}$.
  \end{prop}
 
 The above complexes $\mathcal{C}_{\Lambda}(H,\mathscr{A})$ and $\mathcal{C}_{\Lambda}(H,\overline \mathscr{A})$ may be repackaged as type-$D$ modules over the algebra $\mathcal{K}\otimes_{\mathbb{F}} \mathcal{K}$. See \cite{ZemBordered}*{Section~8.6}.
  We write  $\mathcal{H}_{\Lambda}^{\mathcal{K}\otimes \mathcal{K}}$
  and $\overline \mathcal{H}_{\Lambda}^{\mathcal{K}\otimes \mathcal{K}}$ for these two surgery complexes.
  
 It is convenient to consider the type-$DA$ versions of the Hopf link complexes, and we set
 \[
 {}_{\mathcal{K}} \mathcal{H}_{\Lambda}^{\mathcal{K}}:= \mathcal{H}_{\Lambda}^{\mathcal{K}\otimes \mathcal{K}}\mathrel{\hat{\betaoxtimes}} {}_{\mathcal{K}| \mathcal{K}} [\mathbb{I}^{\Supset}],
 \]
 where the tensor product is taken on a single algebra factor. We define ${}_{\mathcal{K}} \overline{\mathcal{H}}_{\Lambda}^{\mathcal{K}}$ analogously. 

\section{Endomorphisms of the link surgery hypercube}
\label{sec:homology-action}

In this section, we study several endomorphisms of the link surgery hypercube which are related to the standard $\Lambda^* (H_1(Y)/\mathbb{T}ors)$ action on $\mathit{HF}^-(Y)$. In Section~\mathrm{re}f{sec:algebraic-action-link-surgery}, we study one endomorphism of the surgery cube which is obtained by summing over a subset of the structure maps. In Section~\mathrm{re}f{sec:diagrammatic-action}, we study an endomorphism of $\mathcal{C}_{\Lambda}(L)$ induced by a closed curve $\gamma\subseteq \Sigma$. This morphism is computed by counting holomorphic polygons with certain weights. In Section~\mathrm{re}f{sec:relating-actions}, we relate these actions for meridional $\sigma$-basic systems. We call these the \emph{algebraic} and \emph{diagrammatic} actions, respectively.

In Section~\mathrm{re}f{sec:simplifying-connected-sum-1}, we describe an application of these results to simplify the connected sum formula in certain cases.

\subsection{An algebraic $H_1$-action}
\label{sec:algebraic-action-link-surgery}

We now define an action of $\Lambda^* (H_1(S^3_{\Lambda}(L))/\mathbb{T}ors)$ on $\mathcal{C}_{\Lambda}(L)$. Write $H_1(S^3_{\Lambda}(L))=\mathbb{Z}^\ell/\im \Lambda$, where $\mathbb{Z}^\ell$ denotes the free abelian group generated by the meridians $\mu_i$ of the components of $L$.

 Define an endomorphism $F^{K_i}$ of  $\mathcal{C}_{\Lambda}(L)$ to be the sum of all hypercube structure maps for oriented sublinks $\vec{M}\subseteq L$ which contain $+K_i$. Define $F^{-K_i}$ similarly. We define the action $\mathfrak{A}_{[\mu_i]}$ of $\mu_i=(0,\partialots, 1,\partialots 0)$ to be the map $F^{K_i}$.

 Note that $F^{K_i}\simeq F^{-K_i}$ as endomorphisms of $\mathcal{C}_{\Lambda}(L)$, since a chain homotopy is given by projecting to the codimension 1 subcube with $K_i$-coordinate $0$.

 For a general $\gamma\in H_1(S^3_\Lambda(L))$, write $[\gamma]=a_1 \cdot [\mu_1]+\cdots
+ a_\ell \cdot [\mu_\ell]$ and define
\betaegin{equation}
\mathfrak{A}_{[\gamma]}:=a_1 \cdot F^{K_1}+\cdots +a_\ell\cdot F^{K_\ell}.\label{eq:hypercube-homology-action}
\end{equation}
We now prove that this definition gives a well-defined action of $H_1(S^3_{\Lambda}(L))/\mathbb{T}ors$ on $\mathcal{C}_{\Lambda}(L)$.

\betaegin{lem}\label{lem:algebraic-homology-action-link-surgery} The action $\mathfrak{A}_{[\gamma]}$ of $\mathbb{Z}^\ell$ on $\mathcal{C}_{\Lambda}(L)$ descends to an action of $H_1(S^3_\Lambda(L))/\mathbb{T}ors$ which is well-defined up to $\mathbb{F}[\hspace{-.5mm}[ U_1,\partialots, U_\ell]\hspace{-.5mm}] $-equivariant chain homotopy.
\end{lem}
\betaegin{proof}We recall that $H_1(S^3_{\Lambda}(L))$ is isomorphic to $\mathbb{Z}^\ell/\im \Lambda$, where we view $\Lambda$ as the $\ell\times \ell$ symmetric framing matrix for $L$ whose diagonal entries consist of the framings, and whose off diagonal entries consist of the linking numbers between components of $L$. Note that $\gamma\in \mathbb{Z}^\ell$ becomes 0 in $H_1(S^3_\Lambda(L))/\mathbb{T}ors$ if and only if $N\cdot \gamma\in \im \Lambda$ for some $N\in \mathbb{Z}$.

Suppose that $\gamma\in \mathbb{Z}^\ell$ and $[\gamma]=0\in H_1(S^3_{\Lambda}(L))/\mathbb{T}ors$. We will construct a null-homotopy of the map $\mathfrak{A}_{[\gamma]}$ from Equation~\eqref{eq:hypercube-homology-action} as follows. If $\ve{s}\in \mathbb{H}(L)$ and $\varepsilon\in \mathbb{E}_\ell$, write $\mathcal{C}_{\varepsilon}(\ve{s})\subseteq \mathcal{C}_{\varepsilon}\subseteq \mathcal{C}_{\Lambda}(L)$ for the subspace in Alexander grading $\ve{s}$. We will construct a function $\omega_\gamma\colon \mathbb{H}(L)\times \mathbb{E}_\ell\to \mathbb{Z}$ and define a null-homotopy of $\mathfrak{A}_{[\gamma]}$ via the formula:
\[
H_\gamma(\ve{x})=\omega_\gamma(\ve{s},\varepsilon)\cdot \ve{x}
\]
whenever $\ve{x}\in \mathcal{C}_{\varepsilon}(\ve{s})$. 

It suffices to show that we may pick a function $\omega_\gamma\colon \mathbb{H}(L)\times \mathbb{E}_\ell\to \mathbb{Z}$ satisfying 
\betaegin{equation}
\omega_\gamma(\ve{s}, \varepsilon+e_i)=a_{i}+\omega_\gamma(\ve{s},\varepsilon)\quad \text{and} \quad \omega_\gamma(\ve{s}+\Lambda_{i},\varepsilon+e_i)=\omega_\gamma(\ve{s},\varepsilon). 
\label{eq:requirements-omega}
\end{equation}
If such a function $\omega_{\gamma}$ exists, then it is straightforward to verify that 
\[
\mathfrak{A}_{[\gamma]}=[\partial, H_\gamma],
\]
as endomorphisms of $\mathcal{C}_{\Lambda}(L)$.

To establish the existence of a function $\omega_\gamma$ satisfying Equation~\eqref{eq:requirements-omega}, note that we may instead construct a function $\eta_\gamma\colon \mathbb{H}(L)\to \mathbb{Z}$ satisfying
\betaegin{equation}
\eta_\gamma(\ve{s}+\Lambda_i)=\eta_\gamma(\ve{s})-a_i. \label{eq:coherence-eta}
\end{equation}
Given such an $\eta_\gamma$, if $\varepsilon=(\varepsilon_1,\partialots, \varepsilon_{\ell})\in \mathbb{E}_\ell$ we set
\[
\omega_\gamma(\ve{s}, \varepsilon)=\eta_\gamma(\ve{s})+\varepsilon_1\cdot a_1+\cdots +\varepsilon_\ell \cdot a_\ell.
\]
To construct such an $\eta_\gamma$, we pick representatives of each class $\mathbb{H}(L)/\im \Lambda$ and define $\eta_\gamma$ arbitrarily on these elements. We extend $\eta_\gamma$ to all of $\mathbb{H}(L)$ using Equation~\eqref{eq:coherence-eta}. To see that the resulting map $\eta_\gamma$ is well-defined, it suffices to show that if 
\betaegin{equation}
j_1 \Lambda_1+\cdots+j_\ell \Lambda_\ell=0,\label{eq:sum-j-Lambda}
\end{equation}
 then
 \[
 j_1 a_1+\cdots+ j_\ell a_\ell=0.
 \] 
Equation~\eqref{eq:sum-j-Lambda} implies that $(j_1,\partialots, j_\ell)$ is in the null-space of $\Lambda$, so in particular it will vanish when dotted with $(a_1,\partialots,a_\ell)^T$, which is in the rational image of $\Lambda$. The proof is complete.
\end{proof}

\betaegin{rem} It is also straightforward to verify that the above action descends to an action of $\Lambda^* \left(H_1(S^3_{\Lambda}(L))/\mathbb{T}ors\right)$. This may be verified by noting that for each $i$ we have $\mathfrak{A}_{[\mu_i]}^2=0$, and if $i\neq j$ then $[\mathfrak{A}_{[\mu_i]},\mathfrak{A}_{[\mu_j]}]=0$.
\end{rem}

\betaegin{rem}
\label{rem:subcube-refinement}
It is helpful to have the following refinement of Lemma~\mathrm{re}f{lem:algebraic-homology-action-link-surgery} for subcubes of $\mathcal{C}_{\Lambda}(L)$. Suppose that $L$ is partitioned as $L_0\cup L_1$ and let $\varepsilon\in \mathbb{E}_{\ell_0}$ be a fixed coordinate, where $\ell_i=|L_i|$. Consider the subcube $\mathcal{C}_{\Lambda_0}^{(*,\varepsilon)}(L_0;L_1)\subseteq \mathcal{C}_{\Lambda}(L)$ generated by complexes at points $(\nu,\varepsilon)\in \mathbb{E}_{\ell_0}\times \mathbb{E}_{\ell_1}$ where $\varepsilon$ is our chosen coordinate (above) and $\nu$ is any coordinate. Here $\Lambda_0$ is the restriction of $\Lambda$ to $L_0$. The complex $\mathcal{C}_{\Lambda_0}^{(*,\varepsilon)}(L_0;L_1)$ is a module over $\mathbb{F}[\hspace{-.5mm}[ \mathscr{U}_{\ell_0+1},\mathscr{V}_{\ell_0+1},\partialots, \mathscr{U}_{\ell_0+\ell_1},\mathscr{V}_{\ell_0+\ell_1}]\hspace{-.5mm}] $ (the variables for $L_1$). Furthermore, by \cite{ZemBordered}*{Lemma~7.7}, the differential on $\mathcal{C}_{\Lambda_0}^{(*,\varepsilon)}(L_0;L_1)$ commutes with the action of $\mathbb{F}[\hspace{-.5mm}[ \mathscr{U}_{\ell_0+1},\mathscr{V}_{\ell_0+1},\partialots, \mathscr{U}_{\ell_0+\ell_1},\mathscr{V}_{\ell_0+\ell_1}]\hspace{-.5mm}] $.  If $\gamma\in \mathbb{Z}^{\ell_0}$, then we may define an endomorphism $\mathfrak{A}_{[\gamma]}$ on $\mathcal{C}_{\Lambda_0}^{(*,\varepsilon)}(L_0;L_1)$ using  Equation~\eqref{eq:hypercube-homology-action}. Similarly to Lemma~\mathrm{re}f{lem:algebraic-homology-action-link-surgery}, the map $\mathfrak{A}_{[\gamma]}$ gives an action of $H_1(S^3_{\Lambda_0}(L_0))/\mathbb{T}ors$ which is well-defined up to $\mathbb{F}[\hspace{-.5mm}[ \mathscr{U}_{\ell_0+1},\mathscr{V}_{\ell_0+1},\partialots, \mathscr{U}_{\ell_0+\ell_1},\mathscr{V}_{\ell_0+\ell_1}]\hspace{-.5mm}] $-equivariant chain homotopy. To ensure the $\mathbb{F}[\hspace{-.5mm}[ \mathscr{U}_{\ell_0+1},\mathscr{V}_{\ell_0+1},\partialots, \mathscr{U}_{\ell_0+\ell_1},\mathscr{V}_{\ell_0+\ell_1}]\hspace{-.5mm}] $-equivariance of the homotopy, we add to Equation~\eqref{eq:requirements-omega} the requirement that if $e_i\in \mathbb{Z}^{\ell_0+\ell_1}$ is a unit vector pointing in the direction for $K_i\subseteq L_1$, then 
\[
\omega_\gamma(\ve{s}\pm e_i,\varepsilon)=\omega_\gamma(\ve{s}, \varepsilon).
\]
Since $\mathscr{U}_i$ and $\mathscr{V}_i$ have Alexander grading $\pm e_i$, this ensures that our chain homotopies will be $\mathbb{F}[\hspace{-.5mm}[ \mathscr{U}_{\ell_0+1},\mathscr{V}_{\ell_0+1},\partialots, \mathscr{U}_{\ell_0+\ell_1},\mathscr{V}_{\ell_0+\ell_1}]\hspace{-.5mm}] $-equivariant. Noting that there is an affine isomorphism $\mathbb{H}(L)/\mathbb{Z}^{\ell_1}\cong \mathbb{H}(L_0)$ (where $\mathbb{Z}^{\ell_1}$ acts by the meridians of the components of $L_1$), the proof in the absolute case goes through without change.
\end{rem}

\subsection{A Heegaard diagrammatic $H_1$-action}
\label{sec:diagrammatic-action}
In this section, we recall from  \cite{ZemBordered}*{Section~13.3} and \cite{HHSZNaturality}*{Section~6.2} a Heegaard diagrammatic $H_1$-action of a curve $\gamma\subseteq \Sigma$ on the link surgery formula. This description parallels the construction of an $H_1/\mathbb{T}ors$ action on Heegaard Floer homology from \cite{OSDisks}*{Section~4.2.5}. 

 Suppose $L\subseteq S^3$ is a framed link with a system of arcs $\mathscr{A}$ for $L$.  Let $\mathscr{H}$ be a $\sigma$-basic system of Heegaard diagrams for $(L,\mathscr{A})$. Let $\Sigma$ be the underlying Heegaard surface of $\mathscr{H}$. Let $\gamma\subseteq \Sigma$ be a closed curve. We define an endomorphism
\[
\mathcal{A}_{\gamma}\colon \mathcal{C}_{\Lambda}(L,\mathscr{A})\to \mathcal{C}_{\Lambda}(L,\mathscr{A}).
\]
The endomorphism $\mathcal{A}_\gamma$ is induced by a type-$D$ endomorphism  $\mathcal{A}_{\gamma}^1$ of $\mathcal{X}_{\Lambda}(L,\mathscr{A})^{\mathcal{L}}$.

The construction of $\mathcal{A}_\gamma$ is as follows. We assume, for simplicity, that $\gamma$ is represented by a closed 1-chain on $\Sigma$ which is disjoint from the arcs $\mathscr{A}$. We also assume that the arcs $\mathscr{A}$ are embedded in $\Sigma$. The hypercube $\mathcal{C}_{\Lambda}(L,\mathscr{A})$ is built from an $|L|$-dimensional hyperbox of chain complexes. Each constituent hypercube is obtained by pairing two hypercubes of attaching curves, of combined total dimension at most $|L|$, and then extending the remaining axis directions by canonical diffeomorphism maps for surface isotopes of $\Sigma$ which push basepoints along subarcs of the curves in $\mathscr{A}$.

Suppose a subcube of this hyperbox is formed by pairing hypercubes of attaching curves $\mathcal{L}_{\alpha}$ and $\mathcal{L}_{\beta}$, of dimension $n$ and $m$, respectively. The homology action
\[
\mathcal{A}_{\gamma}\colon \ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha},\mathcal{L}_{\beta})\to \ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha},\mathcal{L}_{\beta})
\]
is defined similarly to the hypercube differential in Equation~\eqref{eq:hypercube-differential}, except that a holomorphic polygon representing a class $\ve{p}i$ is weighted by a factor of
\[
\sum_{\alpha\in \mathcal{L}_{\alpha}} \#(\partial_{\alpha}(\ve{p}i)\cap \gamma).
\]
Here, if $\ve{\alphalpha}\in \mathcal{L}_{\alpha}$, then $\partial_{\alpha}(\phi)$ denotes the subset of the boundary of the domain of $\phi$ which lies on $\ve{\alphalpha}$. The homology action $\mathcal{A}_{\gamma}$ on the link surgery hypercube $\mathcal{C}_{\Lambda}(L,\mathscr{A})$ is obtained by performing the above construction to each constituent hypercube of the link surgery formula, and modifying weights of the variables similarly to the construction of the link surgery formula in Section~\mathrm{re}f{sec:basic-systems}.

\subsection{Relating the actions}
\label{sec:relating-actions}

In this section, we relate the two actions $\mathcal{A}_{\gamma}$ and $\mathfrak{A}_{[\gamma]}$ constructed in the previous sections. Our main result is the following:

\betaegin{prop}
\label{prop:homology-action-description}
 Consider a meridional $\sigma$-basic system of Heegaard diagrams for a link $L\subseteq S^3$ and a system of arcs $\mathscr{A}$ for $L$, where each arc is either alpha-parallel or beta-parallel. Let $\Sigma$ be the underlying Heegaard surface.
\betaegin{enumerate}
\item Suppose that $\gamma\subseteq \Sigma$ is a closed curve which is disjoint from the arcs of $\mathscr{A}$. The endomorphism $\mathcal{A}_\gamma$ of $\mathcal{C}_{\Lambda}(L,\mathscr{A})$ satisfies
\[
\mathcal{A}_\gamma\simeq \sum_{i=1}^\ell a_i\cdot \mathcal{A}_{\mu_{i}}
\]
where each $\mu_{i}$ is parallel to the canonical meridian of the component $K_i$ on the diagram $\mathscr{H}$, and $a_i\in \mathbb{F}_2$.
\item If $K_i$ is a component of $L$ and $\varepsilon\in \{0,1\}$, write $\mathcal{C}^{(*,\varepsilon)}_{\Lambda}(L;K_i)$ for the subcube of $\mathcal{C}_{\Lambda}(L)$ generated by complexes at cube points in $\mathbb{E}_{\ell}$ which have $K_i$-coordinate $\varepsilon$.  Embed $K_i$ on $\Sigma$ as a knot trace of the Heegaard link diagram (i.e. the component of $K_i\setminus \{w_i,z_i\}$ oriented from $z_i$ to $w_i$ is disjoint from the beta-curves, and the subarc  oriented from $w_i$ to $z_i$ is disjoint from the alpha curves; we may assume this holds for all diagrams used in the construction of $\mathcal{C}^{(*,\varepsilon)}_{\Lambda}(L;K_i)$). Then, as endomorphisms of $\mathcal{C}^{(*,\varepsilon)}_{\Lambda}(L;K_i)$ we have
\[
\mathcal{A}_{K_i}\simeq \sum_{j\neq i} \mathit{lk}(K_i,K_j) \mathcal{A}_{\mu_{i}}. 
\]
\item For each component $K_i$ of $L$, there is a chain homotopy
\[
\mathcal{A}_{\mu_{i}}\simeq \mathfrak{A}_{[\mu_{i}]}
\]
as endomorphisms of $\mathcal{C}_\Lambda(L)$.
\end{enumerate}
Claims (1) and (3) hold on the level of type-$D$ endomorphisms of $\mathcal{X}_{\Lambda}(L)^{\mathcal{L}}$. 
\end{prop}

We begin with a preliminary lemma:

\betaegin{lem}[\cite{ZemBordered}*{Lemma~13.16}]
\label{lem:homology-action-homotopy-hypercube}
Suppose that $\mathcal{L}_{\alpha}$ and $\mathcal{L}_{\beta}$ are hypercubes of attaching curves on $(\Sigma,\ve{w},\ve{z})$ and that $\gamma$ is a closed 1-chain on $\Sigma$. Suppose that $C\subseteq \Sigma$ is an integral 2-chain such that $\partial C=\gamma+S_{\alpha}+S_{\beta}$ where $S_{\alpha}$ are closed 1-chains which are disjoint from all curves in $\mathcal{L}_{\alpha}$ and $S_{\beta}$ are closed 1-chains which are disjoint from all curves in $\mathcal{L}_{\beta}$. As an endomorphism of $\ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha},\mathcal{L}_{\beta})$ we have
\[
\mathcal{A}_\gamma\simeq 0.
\]
\end{lem}
\betaegin{proof} The proof is given in \cite{ZemBordered}*{Lemma~13.16}, though we repeat it for the benefit of the reader. We construct the following diagram to realize the chain homotopy:
\[
\betaegin{tikzcd} \ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha},\mathcal{L}_{\beta})
	\mathrm{a.r.}[r, "\mathcal{A}_{\gamma}"]
	\mathrm{a.r.}[d, "\id"]
	\mathrm{a.r.}[dr, "H_C",dashed]
& \ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha},\mathcal{L}_{\beta})
	\mathrm{a.r.}[d, "\id"]
\\
\ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha},\mathcal{L}_{\beta})
	\mathrm{a.r.}[r, "0"]&
\ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha},\mathcal{L}_{\beta})
\end{tikzcd}
\]
We define the map $H_C$ to have only length 2 chains in the above diagram, and to send $\ve{x}\in \mathbb{T}_{\alpha_\varepsilon}\cap \mathbb{T}_{\beta_{\nu}}$ to  $n_{\ve{x}}(C)\cdot \ve{x}$, and to be $\mathbb{F}[\hspace{-.5mm}[ U_1,\partialots, U_\ell]\hspace{-.5mm}] $-equivariant. Here $n_{\ve{x}}(C)\in \mathbb{F}$ denotes the intersection number of $C\subseteq \Sigma$ with $\ve{x}$, viewed as sum of $g$-points in $\Sigma$. 

We claim that the hypercube relations are satisfied. To see this, we argue as follows. The relations are equivalent to the following equation:
\betaegin{equation}
H_C\circ D_{(\nu,\varepsilon),(\nu',\varepsilon')}+ D_{(\nu,\varepsilon),(\nu',\varepsilon')}\circ H_C=(\mathcal{A}_\gamma)_{(\nu,\varepsilon),(\nu',\varepsilon')}
\label{eq:homology-action-homotopy-hypercube}
\end{equation}
The maps $D_{(\nu,\varepsilon),(\nu',\varepsilon')}$ and $(\mathcal{A}_\gamma)_{(\nu,\varepsilon),(\nu',\varepsilon')}$ decompose over pairs of increasing sequences
\[
\varepsilon=\varepsilon_1<\cdots<\varepsilon_n=\varepsilon'\quad \text{and} \quad \nu=\nu_1<\cdots<\nu_m=\nu'.
\]
To prove Equation~\eqref{eq:homology-action-homotopy-hypercube}, we observe that if 
\[
\ve{p}i\in \pi_2(\mathbb{T}heta_{\nu_n,\nu_{n-1}},\partialots, \mathbb{T}heta_{\nu_2,\nu_1} ,\ve{x},\mathbb{T}heta_{\varepsilon_1,\varepsilon_2},\partialots ,\mathbb{T}heta_{\varepsilon_{m-1},\varepsilon_m},\ve{y})
\]
 is a class of $(n+m)$-gons, then
\[
0\equiv\#\partial(\partial_{\alpha}(\ve{p}i)\cap C)\equiv (n_{\ve{x}}(C)+n_{\ve{y}}(C))+\#\partial_{\alpha}(\ve{p}i)\cap (\gamma+S_{\alpha}+S_{\beta}) \pmod 2,
\]
We first note that $\partial_{\alpha}(\ve{p}i)\cap S_{\alpha}=\partial_{\beta}(\ve{p}i)\cap S_{\beta}=\emptyset$. Furthermore $\partial_{\alpha}(\ve{p}i)$ and $\partial_{\beta}(\ve{p}i)$ are homologous via the domain of the class $\ve{p}i$ (an integal 2-chain $D(\ve{p}i)$ on $\Sigma$), so we also have $\# (\partial_{\alpha}(\ve{p}i)\cap S_{\beta})\equiv 0$. Hence
\[
n_{\ve{x}}(C)+n_{\ve{y}}(C)\equiv \#(\partial_{\alpha}(\ve{p}i)\cap \gamma)\pmod 2.
\]
This implies Equation~\eqref{eq:homology-action-homotopy-hypercube}, completing the proof.
\end{proof}

By applying Lemma~\mathrm{re}f{lem:homology-action-homotopy-hypercube} to each constituent hypercube of the hyperbox used in the construction of the link surgery formula, we obtain the following corollary concerning the link surgery formula:

\betaegin{cor} Suppose we pick a $\sigma$-basic system of Heegaard diagrams for $(L,\mathscr{A})$ with underlying Heegaard surface $\Sigma$, and $\gamma\subseteq \Sigma$ is a closed curve. Suppose that there is an integral 2-chain $C$ on $\Sigma$ such that $\gamma=\partial C+S_{\alpha}+S_{\beta}$, where $S_{\alpha}$ are closed 1-chains on $\Sigma$ which are disjoint from all alpha curves, and $S_{\beta}$ are closed 1-chains disjoint from all beta curves. Then $\mathcal{A}_{\gamma}\simeq 0$ as endomorphisms of $\mathcal{C}_{\Lambda}(L,\mathscr{A})$ and $\mathcal{X}_{\Lambda}(L,\mathscr{A})^{\mathcal{L}}$.
\end{cor}

We now prove the main result of this section: 

\betaegin{proof}[Proof of Proposition~\mathrm{re}f{prop:homology-action-description}]
We begin with the first claim. We use a meridional $\sigma$-basic system, as described in Section~\mathrm{re}f{sec:basic-systems}. Let $(\Sigma,\ve{\alphalpha},\ve{\betaeta},\ve{w},\ve{z})$ denote the underlying Heegaard link diagram for $(S^3,L)$. Write $(\Sigma,\ve{\alphalpha}_0,\ve{\betaeta}_0)$ for the partial Heegaard diagram obtained by deleting the special alpha and beta curves (i.e. the meridians of the components of $L$). If we attach compressing disks along the $\ve{\alphalpha}_0$ and $\ve{\betaeta}_0$ curves, and fill any 2-sphere boundary components with 3-balls, we obtain $S^3\setminus \nu (L)$. By including $\Sigma$ into $S^3\setminus \nu(L)$ in this way, we may view $\gamma$ as being in $S^3\setminus \nu(L)$. We write $\mu_1,\partialots, \mu_\ell\subseteq \Sigma$ for curves which are parallel to the special meridional alpha and beta curves. The meridians $\mu_1,\partialots, \mu_\ell$  generate $H_1(S^3\setminus \nu(L))$. Hence, $\gamma$ is homologous in $S^3\setminus \nu(L)$ to a linear combination of these meridians, i.e. we may write
\[
\gamma-a_1\cdot \mu_1-\cdots -a_\ell \cdot\mu_\ell=\partial C,
\]
for some 2-chain $C$ in $S^3\setminus \nu(L)$ and some integers $a_1,\partialots, a_\ell$. Such a relation induces a 2-chain $C'$ on $(\Sigma,\ve{\alphalpha}_0,\ve{\betaeta}_0)$ such that $\partial C'$ is $\gamma-a_{1}\cdot \mu_1-\cdots -a_\ell\cdot \mu_\ell+ S_{\alpha}+ S_{\beta}$, where $S_{\alpha}$ are parallel copies of curves in $\ve{\alphalpha}_0$ and $\ve{\betaeta}_0$, and $\mu_i$ are the canonical meridians. In particular, $S_{\alpha}$ is disjoint from all alpha curves in the $\sigma$-basic system and $S_{\beta}$ is disjoint from all beta curves in the $\sigma$-basic system.  In our $\sigma$-basic system, the curves $\ve{\alphalpha}_0$ and $\ve{\betaeta}_0$ appear via small Hamiltonian translates in every collection of alpha or beta curves.  The first claim of the theorem statement now follows from Lemma~\mathrm{re}f{lem:homology-action-homotopy-hypercube}.

We now consider the second claim. Assume for concreteness that the canonical meridian of $K_i$ is an alpha curve $\alpha_i^s$ (i.e. the arc for $K_i$ is beta-parallel). We observe that the above construction gives an embedding of $\Sigma$ into the complement of $L$, and the quantity $a_i$ is the linking number $\mathit{lk}(\gamma, K_i)$. When we restrict to the subcube  $\mathcal{C}^{(*,\varepsilon)}_{\Lambda}(L;K_i)$, we may now view the curve $\gamma$ as being embedded $S^3\setminus\nu(L-K_i)$, since the canonical meridian of $K_i$ is stationary in this subcube. In particular, the trace of the knot $K_i$ on $\Sigma$ will be homologous on $\Sigma$ to $\sum_{i\neq j} \mathit{lk}(K_j,K_i) [\mu_j]$ and a sum of curves on $\Sigma$ which are small translates of the curves in $\ve{\alphalpha}_0\cup \{\alpha_i^s\}$ or $\ve{\betaeta}_0$. The claim now follows from Lemma~\mathrm{re}f{lem:homology-action-homotopy-hypercube}.

We now prove the third claim by constructing a homotopy $\mathcal{A}_{\mu_i}\simeq F^{-K_i}$. We observe that $F^{-K_i}\simeq F^{K_i}$, which is by definition the same as $\mathfrak{A}_{[\mu_i]}$.  Write $\mathcal{L}_{\alpha}$ and $\mathcal{L}_{\beta}$ for the hyperboxes of attaching curves from our $\sigma$-basic system. We suppose $K_i\subseteq L$ is a link component which is beta-parallel. We decompose the $K_i$-direction of $\mathcal{L}_{\alpha}$ as
\[
\betaegin{tikzcd}
\mathcal{L}_{\alpha_1}
\mathrm{a.r.}[r]& \mathcal{L}_{\alpha_2}\mathrm{a.r.}[r] &\cdots \mathrm{a.r.}[r]& \mathcal{L}_{\alpha_n}.
\end{tikzcd}
\]
We assume each arrow is either a morphism of hyperboxes of attaching curves (i.e. a collection of Floer chains satisfying the compatibility relations) or is the canonical surface isotopy which moves $z_i$ to $w_i$ along the short path connecting them on the Heegaard diagram. 

As described in Section~\mathrm{re}f{sec:basic-systems}, the hypercube $\mathcal{C}_{\Lambda}(L)$ is completely determined by the compression of the hypercube
\[
\betaegin{tikzcd}
\ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha_1},\mathcal{L}_{\beta})\mathrm{a.r.}[r, "F_{1,2}"]& \ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha_2},\mathcal{L}_\beta) \mathrm{a.r.}[r, "F_{2,3}"]&\cdots\mathrm{a.r.}[r, "F_{n-1,n}"]& \ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha_n},\mathcal{L}_{\beta}).
\end{tikzcd}
\]
For convenience, write $\mathcal{C}_i$ for the hyperbox $\ve{\mathbb{C}F}^-(\mathcal{L}_{\alpha_i},\mathcal{L}_{\beta})$.

Let $\mu_i'$ be a translation on the Heegaard diagram of $\mu_i$.
We observe that if $C\subseteq \Sigma$ is a 2-chain whose boundary is $\mu_i-\mu_i'$, then the following hyperbox compresses horizontally to the identity map on $\mathbb{C}one(\mathcal{A}_{\mu_i})$:
\betaegin{equation}
\betaegin{tikzcd}[labels=description, row sep=1cm, column sep=1cm]
\mathcal{C}_n
	\mathrm{a.r.}[dr, dashed, "H_C"]
	\mathrm{a.r.}[r, "\id"]
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i}"]
&
\mathcal{C}_n
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i'}"]
	\mathrm{a.r.}[dr,dashed, "H_C"]
	\mathrm{a.r.}[r, "\id"]
&
\mathcal{C}_n
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i}"]
\\
\mathcal{C}_n
	\mathrm{a.r.}[r, "\id"]
&
\mathcal{C}_n
	\mathrm{a.r.}[r, "\id"]
&
\mathcal{C}_n
\end{tikzcd}
\label{eq:move-move-back}
\end{equation}

Next, we observe that for any $k\in \{1,\partialots, n-1\}$, the horizontal compressions of the following hyperboxes are homotopic as hyperbox morphisms from $\mathbb{C}one(\mathcal{A}_{\mu_i}\colon \mathcal{C}_k\to \mathcal{C}_k)$ and $\mathbb{C}one(\mathcal{A}_{\mu_i'}\colon \mathcal{C}_{k+1}\to \mathcal{C}_{k+1})$:
\betaegin{equation}
\betaegin{tikzcd}[labels=description,column sep=1.4cm, row sep=1.2cm]
\mathcal{C}_k
	\mathrm{a.r.}[dr, dashed, "F_{k,k+1}^\gamma"]
	\mathrm{a.r.}[r, "F_{k,k+1}"]
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i}"]
&
\mathcal{C}_{k+1}
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i}"]
	\mathrm{a.r.}[dr,dashed, "H_C"]
	\mathrm{a.r.}[r, "\id"]
&
\mathcal{C}_{k+1}
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i'}"]
\\
\mathcal{C}_k
	\mathrm{a.r.}[r, "F_{k,k+1}"]
&
\mathcal{C}_{k+1}
	\mathrm{a.r.}[r, "\id"]
&
\mathcal{C}_{k+1}
\end{tikzcd}
\quad \text{and} \quad
\betaegin{tikzcd}[labels=description, column sep=1.4cm, row sep=1.2cm]
\mathcal{C}_k
	\mathrm{a.r.}[dr, dashed, "H_C"]
	\mathrm{a.r.}[r, "\id"]
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i}"]
&
\mathcal{C}_{k+1}
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i'}"]
	\mathrm{a.r.}[dr,dashed, "F_{k,k+1}^{\mu_i}"]
	\mathrm{a.r.}[r, "F_{k,k+1}"]
&
\mathcal{C}_{k+1}
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i'}"]
\\
\mathcal{C}_k
	\mathrm{a.r.}[r, "\id"]
&
\mathcal{C}_{k+1}
	\mathrm{a.r.}[r, "F_{k,k+1}"]
&
\mathcal{C}_{k+1}
\end{tikzcd}
\label{eq:two-compressions}
\end{equation}
The above claim follows from Lemma~\mathrm{re}f{lem:homology-action-homotopy-hypercube}, which constructs a hyperbox which realizes a chain homotopy $\mathcal{A}_{\mu_i}\simeq \mathcal{A}_{\mu_i'}$ as endomorphisms of $\mathbb{C}one(F_{j,j+1})$. This hyperbox realizes exactly the chain homotopy between the compressions described above.

In particular, from the two results related to Equations~\eqref{eq:move-move-back} and ~\eqref{eq:two-compressions}, we conclude that the hypercube map $\mathcal{A}_{\mu_i}\colon \mathcal{C}_{\Lambda}(L)\to \mathcal{C}_{\Lambda}(L)$ can be described as the compression of the following hyperbox, for any $k$:
\[
\betaegin{tikzcd}[column sep=.6cm, row sep=1cm]
\mathcal{C}_1
	\mathrm{a.r.}[r, "F_{1,2}"]
	\mathrm{a.r.}[dr,dashed]
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i}",labels=description]
& 
\mathcal{C}_2
	\mathrm{a.r.}[r, "F_{2,3}"]
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i}",labels=description]
	\mathrm{a.r.}[dr,dashed]
&
\cdots
\mathrm{a.r.}[r]
&
\mathcal{C}_k
	\mathrm{a.r.}[r, "\id"]
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i}",labels=description]
	\mathrm{a.r.}[dr,dashed, "H_C",labels=description]
& 
\mathcal{C}_k
	\mathrm{a.r.}[r,"F_{k,k+1}"]
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i'}",labels=description]
	\mathrm{a.r.}[dr,dashed]
&
\mathcal{C}_{k+1}
	\mathrm{a.r.}[r]
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i'}",labels=description]
&
\cdots 
\mathrm{a.r.}[r]
&
\mathcal{C}_n
	\mathrm{a.r.}[r, "\id"]
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i'}",labels=description]
	\mathrm{a.r.}[dr, dashed,"H_C",labels=description]
&\mathcal{C}_n
	\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i}",labels=description]
\\
\mathcal{C}_1\mathrm{a.r.}[r, "F_{1,2}"]& \mathcal{C}_2\mathrm{a.r.}[r, "F_{2,3}"]& \cdots\mathrm{a.r.}[r]& \mathcal{C}_k\mathrm{a.r.}[r, "\id"]& \mathcal{C}_k\mathrm{a.r.}[r,"F_{k,k+1}"]&\mathcal{C}_{k+1}\mathrm{a.r.}[r]& \cdots \mathrm{a.r.}[r] &\mathcal{C}_n\mathrm{a.r.}[r, "\id"]& \mathcal{C}_n
\end{tikzcd}
\]
We may pick $C$, $\mu_i$, $\mu_i'$ and $k$ so that $\mu_i$ is disjoint from $\mathcal{L}_{\alpha_1},\partialots, \mathcal{L}_{\alpha_{k}}$, and $\mu_i'$ is disjoint from $\mathcal{L}_{\alpha_{k+1}},\partialots, \mathcal{L}_{\alpha_n}$, and $C$ covers all of the special meridional curve of $\mathcal{L}_{\alpha_k}$ (and in particular, $n_C(p)\equiv 1$ for each $p$ in the special meridianal curve of any curve collection in $\mathcal{L}_{\alpha_k}$). We assume, additionally, that $C$ is disjoint from the curves of $\mathcal{L}_{\alpha_n}$ (in particular, the initial special meridional curve $\alpha_i^s$). See Figure~\mathrm{re}f{cor:1}. In particular, the only diagonal map which will be non-trivial in the above compression will be the one with domain $\mathcal{C}_k$. Since $n_C(\ve{x})\equiv 1$ for every intersection point $\ve{x}$ in any of the Floer complexes comprising $\mathcal{C}_k$, the map $H_C$ will act by the identity on $\mathcal{C}_k$.  In particular, the compression coincides with the diagram
\[
\betaegin{tikzcd}\mathcal{C}_{\Lambda}(L)\mathrm{a.r.}[d, "\mathcal{A}_{\mu_i}"] \\ \mathcal{C}_{\Lambda}(L)
\end{tikzcd}
\simeq
\betaegin{tikzcd}[labels=description, column sep=2cm]
\mathcal{C}^{(*,0)}(L)\mathrm{a.r.}[d, "0"]\mathrm{a.r.}[r, "F^{-K_i}"]\mathrm{a.r.}[dr,dashed, "F^{-K_i}"]& \mathcal{C}^{(*,1)}(L)\mathrm{a.r.}[d, "0"]\\
\mathcal{C}^{(*,0)}(L)\mathrm{a.r.}[r, "F^{-K_i}"]& \mathcal{C}^{(*,1)}(L)
\end{tikzcd}.
\]
The vertical direction is the homology action, so the proof is complete.
\end{proof}

\betaegin{figure}[ht]
\input{cor1.pdf_tex}
\caption{The curves $\mu_i$, $\mu_i'$ and the special meridian $\alpha_i^s$ of component $J\subseteq L$. The shaded annulus is the 2-chain $C$. The dashed curves labeled $\alpha'$ are the translates of $\alpha_i^s$ appearing in the $\sigma$-basic system.}
\label{cor:1}
\end{figure}

\betaegin{example} We illustrate Proposition~\mathrm{re}f{prop:homology-action-description} for the knot surgery formula. In this case, the result is already known (using different notation). See \cite{OSProperties}*{Theorem~9.23}. Ozsv\'{a}th and Szab\'{o} \cite{OSIntegerSurgeries} proved that
\[
\ve{\mathbb{C}F}^-(S^3_n(K))\cong \mathbb{C}one(v+h_n\colon \mathbb{A}(K)\to \mathbb{B}(K)).
\]
Though we do not need this fact, one can show that the diagrammatic $H_1$-action $\mathcal{A}_{\mu}$ coincides with the ordinary homology action of the meridian of $K$ on $\ve{\mathbb{C}F}^-(S^3_n(K))$. (Of course, this is null-homotopic when $n\neq 0$). Proposition~\mathrm{re}f{prop:homology-action-description} translates to saying that the homology action of the meridian $\mu$ coincides with the endomorphism $h_n$ on the mapping cone complex.
\end{example}

\subsection{Application to the connected sum formula}
\label{sec:simplifying-connected-sum-1}

One important consequence of Proposition~\mathrm{re}f{prop:homology-action-description}, Lemma~\mathrm{re}f{lem:algebraic-homology-action-link-surgery} and Remark~\mathrm{re}f{rem:subcube-refinement} is the following:

\betaegin{cor}
\label{cor:simplify-tensor-product}
 Suppose that $L_1$ and $L_2$ are links in $S^3$ with framings $\Lambda_1$ and $\Lambda_2$, and with systems of arcs $\mathscr{A}_1$ and $\mathscr{A}_2$, respectively. Let $K_1\subseteq L_1$ and $K_2\subseteq L_2$ be distinguished components, whose arcs are alpha-parallel and beta-parallel, respectively. Let $\Lambda_1'$ denote the restriction of the framing $\Lambda_1$ to $L_1-K_1$. Suppose that $K_1\subseteq S^3_{\Lambda_1'}(L_1-K_1)$ is rationally null-homologous. 
Write $\mathcal{X}_{\Lambda_i}(L_i,\mathscr{A}_i)^{\mathcal{K}_i}$ for the type-$D$ modules obtained from $\mathcal{X}_{\Lambda_i}(L_i,\mathscr{A}_i)^{\mathcal{L}_{\ell_i}}$ by boxing ${}_{\mathcal{K}} \mathcal{D}_0$ into each algebra component except for the one corresponding to $K_i$.
Then
\[
\betaegin{split}
&\left(\mathcal{X}_{\Lambda_1}(L_1,\mathscr{A}_1)^{\mathcal{K}_1}, \mathcal{X}_{\Lambda_2}(L_2,\mathscr{A}_2)^{\mathcal{K}_2}\right)\mathrel{\hat{\betaoxtimes}} {}_{\mathcal{K}_1| \mathcal{K}_2} W_l^{\mathcal{K}}
\\
\simeq  &\left(\mathcal{X}_{\Lambda_1}(L_1,\mathscr{A}_1)^{\mathcal{K}_1}, \mathcal{X}_{\Lambda_2}(L_2,\mathscr{A}_2)^{\mathcal{K}_2}\right)\mathrel{\hat{\betaoxtimes}} {}_{\mathcal{K}_1|\mathcal{K}_2} M^{\mathcal{K}}.
\end{split}
\]
\end{cor}
Corollary~\mathrm{re}f{cor:simplify-tensor-product} is most easily proven by using the formulation of the connected sum formula in Equation~\eqref{eq:alternate-pairing-theorem}. The extra term in the differential of the tensor product with $W_l$ (resulting from the $\partialelta_5^1$ term in Equation~\eqref{eq:delta-5-1-merge}) factors through the homology action $\mathcal{A}_{[K_1]}\otimes \id$ on $\mathcal{C}^{(*,1)}(L_1)\otimes \mathcal{C}^{(*,1)}(L_2)$. If $K_1$ is rationally null-homologous in the surgery on the other components, this map will be null-homotopic by Lemma~\mathrm{re}f{lem:algebraic-homology-action-link-surgery}.

\section{Simplifying the pairing theorem}

In this section, we recall the category of \emph{Alexander modules} from \cite{ZemBordered}*{Section~6}. After recalling precise definitions, we will prove the following result, which is essential to our proof of Theorem~\mathrm{re}f{thm:pairing-main} when $b_1(Y(G))>0$. The results of this section are not essential when $b_1(Y(G))=0$.

\betaegin{prop}\label{prop:merge=pair-of-pants}
 Suppose that $\mathcal{X}_1^{\mathcal{K}}$ and $\mathcal{X}_2^{\mathcal{K}}$ are type-$D$ Alexander modules which are homotopy equivalent to finitely generated type-$D$ modules (i.e. have homotopy equivalent models where $\mathcal{X}_i$ are finite dimensional $\mathbb{F}$-vector spaces). Then
 \[
\left( \mathcal{X}_1^{\mathcal{K}},\mathcal{X}_2^{\mathcal{K}}\right)\mathrel{\hat{\betaoxtimes}} {}_{\mathcal{K}| \mathcal{K}} W_l^{\mathcal{K}}\simeq \left( \mathcal{X}_1^{\mathcal{K}}, \mathcal{X}_2^{\mathcal{K}}\right)\mathrel{\hat{\betaoxtimes}} {}_{\mathcal{K}| \mathcal{K}} M^{\mathcal{K}}.
 \]
 The same holds for $W_r$ in place of $W_l$.
\end{prop}

Our proof is inspired by work of the author with Hendricks and Manolescu \cite{HMZConnectedSum}*{Section~6}, where an extra term in an involutive connected sum formula is shown to be null-homotopic. The proof occupies the next several subsections.

\subsection{Finitely generated $\mathcal{K}$-modules}

We now discuss type-$D$ Alexander modules $\mathcal{X}^{\mathcal{K}}$ where $\mathcal{X}$ is finitely generated. Such modules form a subcategory which admits a somewhat simpler description not involving linear topological spaces.

We denote by $\ve{\mathcal{K}}$ the completion of the algebra $\mathcal{K}$ with respect to the topology described in Section~\mathrm{re}f{def:knot-topology}. As a vector space, we have the following isomorphisms:
\[
\betaegin{split}
\ve{I}_0\cdot \ve{\mathcal{K}} \cdot \ve{I}_0&\cong \mathbb{F}[\hspace{-.5mm}[ \mathscr{U},\mathscr{V}]\hspace{-.5mm}] \\
 \ve{I}_1\cdot \ve{\mathcal{K}}\cdot \ve{I}_0&\cong \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}]  [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}]  \langle \tau\rangle \oplus \mathbb{F}[\hspace{-.5mm}[ \mathscr{V},\mathscr{V}^{-1}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}]  \langle \sigma \rangle\\
\ve{I}_1\cdot \ve{\mathcal{K}} \cdot \ve{I}_1&\cong \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] .
\end{split}
\]

The space $\ve{\mathcal{K}}$ is again an algebra, and $\mu_2$ is continuous when viewed as a map from $\ve{\mathcal{K}} \mathrel{\vec{\otimes}}_{\mathrm{I}} \ve{\mathcal{K}}$ to $\ve{\mathcal{K}}$. 

Since $\ve{\mathcal{K}}$ is an algebra, we may consider the category of finitely generated type-$D$ modules over $\ve{\mathcal{K}}$. We write $\MOD^{\ve{\mathcal{K}}}_{\mathit{fg}}$ for this category. These are ordinary type-$D$ modules over $\ve{\mathcal{K}}$ (i.e. not equipped with a topology), which have underlying vector spaces which are finitely generated over $\mathbb{F}$. Morphisms in this category are ordinary type-$D$ morphisms, i.e. maps $f^1\colon \mathcal{X}\to \mathcal{Y}\otimes \ve{\mathcal{K}}$. 

 There is a related category, $\MOD^{\mathcal{K}}_{\mathit{fg},\mathfrak{a}}$, which is the set of type-$D$ Alexander modules over $\mathcal{K}$, which are finitely generated over $\mathbb{F}$.

\betaegin{lem}[\cite{ZemBordered}*{Section~6.7}] The categories $\MOD^{\mathcal{K}}_{\mathit{fg},\mathfrak{a}}$ and $\MOD^{\ve{\mathcal{K}}}_{\mathit{fg}}$ are equivalent.
\end{lem}

The category of finitely generated type-$D$ modules is important for the bordered theory for the following reason. 

\betaegin{prop}[\cite{ZemBordered}*{Proposition~18.11}]
\label{prop:finite-generation}
Suppose $L=K_1\cup \cdots \cup K_n$ is a link in $S^3$ with framing $\Lambda$. Write $\mathcal{X}_{\Lambda}(L_{1,\partialots, n-1}, K_n)^{\mathcal{K}}$ for the type-$D$ module obtained by tensoring $(n-1)$-copies of the type-$A$ module for a solid torus, ${}_{\mathcal{K}} \mathcal{D}_0$,  with $\mathcal{X}_{\Lambda}(L)^{\mathcal{L}_n}$ along the algebra components for $K_1,\partialots, K_{n-1}$. Then $\mathcal{X}_{\Lambda}(L_{1,\partialots, n-1}, K_n)^{\mathcal{K}}$ is homotopy equivalent to a finitely generated type-$D$ module.
\end{prop}

For the sake of exposition, we now explain the definition of a finitely generated type-$D$ module over $\ve{\mathcal{K}}$ in more detail:

\betaegin{lem}
\label{lem:restate-type-D}A finitely generated type-$D$ Alexander module over $\mathcal{K}$ is equivalent to the following data:
\betaegin{enumerate}
\item A pair of finite dimensional $\mathbb{F}$-vector spaces $\mathcal{X}_0$ and $\mathcal{X}_1$, equipped with internal differentials
\[
\partial_0\colon \mathcal{X}_0\to \mathcal{X}_0\otimes \mathbb{F}[\hspace{-.5mm}[ \mathscr{U},\mathscr{V}]\hspace{-.5mm}] \quad \text{and} \quad \partial_1\colon \mathcal{X}_1\to \mathcal{X}_1\otimes \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}][\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] .
\]
\item Maps
\[
\betaegin{split}
v&\colon \mathcal{X}_0\to  \mathcal{X}_1\otimes \mathbb{F}[\hspace{-.5mm}[ \mathscr{V},\mathscr{V}^{-1}][\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] \quad \text{and}
\\ h&\colon \mathcal{X}_0\to \mathcal{X}_1\otimes \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] .
\end{split}
\]
 Furthermore, if we extend $v$ $I$-equivariantly to a map $V$ with domain $\mathcal{X}_0\otimes \mathbb{F}[\hspace{-.5mm}[ \mathscr{U},\mathscr{V}]\hspace{-.5mm}] $, then $V$ is a chain map. If we extend $h$ $T$-equivariantly to a map $H$ with domain $\mathcal{X}_0\otimes \mathbb{F}[\hspace{-.5mm}[ \mathscr{U},\mathscr{V}]\hspace{-.5mm}] $, then $H$ is a chain map. (Recall that $I$ and $T$ are the algebra morphisms in the definition of $\mathcal{K}$; see Section~\mathrm{re}f{sec:surgery-algebra}).
\end{enumerate}
Morphisms between finitely generated type-$D$ modules are similar. As a particularly important special case, suppose $h$ and $h'$ are two maps from $\mathcal{X}_0$ to $\mathcal{X}_1\otimes \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $, whose $T$-equivariant extensions are chain maps, as above. Suppose there is a third map $j\colon \mathcal{X}_0\to \mathcal{X}_1\otimes \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $, and let $J$ be its $T$-equivariant extension to $\mathcal{X}_0\otimes \mathbb{F}[\hspace{-.5mm}[ \mathscr{U},\mathscr{V}]\hspace{-.5mm}] $. If $\partial_1\circ J+J\circ \partial_0=H+H'$, then the type-$D$ module obtained by replacing $h$ with $h'$ is homotopy equivalent to $\mathcal{X}^{\mathcal{K}}$.
\end{lem}

We now prove a key lemma:

\betaegin{lem}\label{lem:classification-PID} Suppose that $\mathcal{X}^{\mathcal{K}}$ is a finitely generated type-$D$ module such that $\mathcal{X}_1\otimes \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $ admits a $\mathbb{Z}_2$-valued grading (with $\mathscr{U}$ and $\mathscr{V}$ viewed as having grading $0\in \mathbb{Z}_2$). The chain complex  $\mathcal{X}_1\otimes \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $ is  chain isomorphic to a direct sum of 1-step complexes (i.e. complexes with a single generator and vanishing differential) and 2-step complexes (i.e. complexes with two generators $\ve{x},\ve{y}$ and $\partial(\ve{x})=\ve{y}\otimes \alpha_{\ve{y},\ve{x}}$ ). Furthermore, in the two step complexes, we may assume each $\alpha_{\ve{y},\ve{x}}$ is of the form  $U^i$ for some $i\in \mathbb{N}$, where $U=\mathscr{U}\mathscr{V}$.
\end{lem}
\betaegin{proof} The key observation is that $\mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] $ is a field, so $\mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $ is a PID. In fact, the ideals of $\mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $ are all of the form $(U^i)$ for $i\gammae 0$. Hence, the proof follows immediately from the classification theorem for finitely generated free chain complexes over a PID. We sketch the argument very briefly in our present setting. One first considers the arrows of the differential on $\mathcal{X}_0\otimes \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $ which are weighted by units. If any such arrow exists, we pick one arbitrarily. Suppose this arrow goes from $\ve{x}$ to $\ve{y}\otimes \alpha$. Since the ideals of $\mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $ are all of the form $(U^i)$, one may perform a change of basis so that there are no other arrows from $\ve{x}$ or to $\ve{y}$. After performing this change of basis, there are also no arrows to $\ve{x}$ or from $\ve{y}$, since $\partialelta^1$ squares to 0. Performing a further change of basis yields the summand consisting of the subcomplex $\partialelta^1(\ve{x})= \ve{y}\otimes 1$. We repeat this until all arrows (except on these two step complexes) have weight in the  $(U^1)$. We repeat the above procedure to isolate arrows with weight in $(U^1)\setminus (U^2)$ until, outside of the isolated 2-step complexes, all arrows have weight in the ideal $(U^2)$. We repeat this procedure until we are left with only 1-step and 2-step complexes. After a chain isomorphism, the weights on the algebra elements in the 2-step complexes may be taken to be powers of $U$.
\end{proof}

\betaegin{cor} \label{cor:eliminate-homology-action}
Let $\mathcal{X}^{\mathcal{K}}$ be a finitely generated type-$D$ module. Consider the endomorphism $\mathcal{A}:=\mathscr{U} \betaPhi+\mathscr{V}\betaPsi$ of the chain complex $\mathcal{X}_1\otimes \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $, where $\betaPhi$ and $\betaPsi$ denote the algebraically defined basepoint actions define in Section~\mathrm{re}f{sec:background}. Then $\mathcal{A}$ is null-homotopic on $\mathcal{X}_1\otimes \mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $ via a $\mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $-equivariant chain homotopy. 
\end{cor}
\betaegin{proof} The maps $\betaPhi$ and $\betaPsi$ commute with homotopy equivalences. Using Lemma~\mathrm{re}f{lem:classification-PID}, it is sufficient to observe that $\mathscr{U}\betaPhi+\mathscr{V}\betaPsi\equiv 0$ for a 1-step complex and for a 2-step complex of the form $\partialelta^1(\ve{x})=\ve{y}\otimes U^i$, since $(\mathscr{U} \partial_{\mathscr{U}}+\mathscr{V} \partial_{\mathscr{V}})(U^i)=0,$ as $U=\mathscr{U}\mathscr{V}$.
\end{proof}

We are now in position to prove the main theorem of this section, Proposition~\mathrm{re}f{prop:merge=pair-of-pants}:

\betaegin{proof}[Proof of Proposition~\mathrm{re}f{prop:merge=pair-of-pants}] 
By Lemma~\mathrm{re}f{lem:restate-type-D}, it is sufficient to show that the corresponding $h$ maps of the two type-$D$ modules are chain homotopic. We observe that by Equation~\eqref{eq:homology-action-basepoint} the difference factors through the endomorphism $(\mathscr{U} \betaPhi_1+\mathscr{V}\betaPsi_1)\otimes \id$ of $\mathcal{C}^{(*,1)}(L_1)\otimes \mathcal{C}^{(*,1)}(L_2)$. Here, $\betaPhi_1$ and $\betaPsi_1$ denote the algebraic basepoint actions of $\mathcal{C}^{(*,1)}(L_1)$. The basepoint actions $\betaPhi_1$ and $\betaPsi_1$ commute with homotopy equivalences up to chain homotopy (cf. \cite{ZemConnectedSums}*{Lemma~2.8}). By Lemma~\mathrm{re}f{lem:restate-type-D}, it is sufficient to show that $(\mathscr{U} \betaPhi_1+\mathscr{V}\betaPsi_1)\otimes \id$ is null-homotopic if we take coefficients in $\mathbb{F}[\mathscr{V},\mathscr{V}^{-1}]\hspace{-.5mm}] [\hspace{-.5mm}[ \mathscr{U}]\hspace{-.5mm}] $. This is a consequence of Corollary~\mathrm{re}f{cor:eliminate-homology-action}, so the proof is complete.
\end{proof}

\betaegin{rem} Our proof can also be used to show that  if $\mathcal{X}^{\mathcal{K}}$ is a type-$D$  Alexander  module which is homotopy equivalent to a finitely generated type-$D$ module, then there is a homotopy equivalence
\[
\mathcal{X}^{\mathcal{K}}\simeq \mathcal{X}^{\mathcal{K}}\mathrel{\hat{\betaoxtimes}} {}_{\mathcal{K}} \mathcal{T}^{\mathcal{K}}.
\]
In the above, ${}_{\mathcal{K}} \mathcal{T}^{\mathcal{K}}$ is the transformer bimodule from \cite{ZemBordered}*{Section~14}. The above equation follows because the difference in the $h$-maps between $\mathcal{X}^{\mathcal{K}}$ and $\mathcal{X}^{\mathcal{K}}\mathrel{\hat{\betaoxtimes}}{}_{\mathcal{K}} \mathcal{T}^{\mathcal{K}}$ may also be factored through the map $\mathscr{U} \betaPhi+\mathscr{V} \betaPsi$. See the proof of \cite{ZemBordered}*{Theorem~14.1}.
\end{rem}

\section{Completing the proof of Theorem~\mathrm{re}f{thm:main}}
\label{sec:proof}

In this section, we describe the proof of Theorem~\mathrm{re}f{thm:main}.

\subsection{Homological perturbation lemma for hypercubes}

In this section, we review a version of the homological perturbation lemma for hypercubes. This lemma is similar to work of Huebschmann-Kadeishvili \cite{HK_Homological_Perturbation}. See  \cite{Liu2Bridge}*{Section~5.6} for a similar though slightly less explicit result for transferring hypercube structure maps along homotopy equivalences.

\betaegin{lem}[\cite{HHSZDuals}*{Lemma~2.10}]\label{lem:homological-perturbation-cubes}
Suppose that $\mathcal{C}=(C_\varepsilon,D_{\varepsilon,\varepsilon'})$ is a hypercube of chain complexes, and $(Z_\varepsilon,\partialelta_\varepsilon)_{\varepsilon\in \mathbb{E}_n}$ is a collection of chain complexes. Furthermore, suppose there are maps 
\[
\pi_{\varepsilon}\colon C_\varepsilon\to Z_\varepsilon\qquad i_{\varepsilon}\colon Z_\varepsilon\to C_\varepsilon \qquad h_\varepsilon\colon C_\varepsilon\to C_\varepsilon,
\]
satisfying
\[
\pi_{\varepsilon}\circ i_{\varepsilon}=\id, \qquad i_{\varepsilon}\circ \pi_{\varepsilon}=\id+[\partial, h_\varepsilon], \quad  h_\varepsilon\circ h_\varepsilon=0, \quad \pi_{\varepsilon}\circ h_{\varepsilon}=0\quad \text{and} \quad h_{\varepsilon}\circ i_{\varepsilon}=0
\]
and such that $\pi_{\varepsilon}$ and $i_{\varepsilon}$ are chain maps.
With the above data chosen, there are canonical hypercube structure maps $\partialelta_{\varepsilon,\varepsilon'}\colon Z_{\varepsilon}\to Z_{\varepsilon'}$ so that $\mathcal{Z}=(Z_\varepsilon,\partialelta_{\varepsilon,\varepsilon'})$ is a hypercube of chain complexes, and also there are morphisms of hypercubes
\[
\betaPi\colon \mathcal{C}\to \mathcal{Z}, \quad I\colon  \mathcal{Z}\to \mathcal{C}\quad \text{and}\quad H\colon \mathcal{C}\to \mathcal{C}
\]
such that
\[
\betaPi\circ I=\id\quad \text{and} \quad I\circ \betaPi= \id+\partial_{\Mor}(H)
\]
and such that $I$ and $\betaPi$ are chain maps.
\end{lem}

The structure maps $\partialelta_{\varepsilon,\varepsilon'}$ and the morphisms $\betaPi$ and $I$ have a concrete formula. We begin with $\partialelta_{\varepsilon,\varepsilon'}$. Suppose that $\varepsilon<\varepsilon'$ are points in $\mathbb{E}_n$.   The hypercube structure maps $\partialelta_{\varepsilon,\varepsilon'}$ are given by the following formula:
\[
\partialelta_{\varepsilon,\varepsilon'}:=\sum_{\varepsilon=\varepsilon_1<\cdots<\varepsilon_j=\varepsilon'}  \pi_{\varepsilon'}\circ D_{\varepsilon_{j-1},\varepsilon_j}\circ h_{\varepsilon_{j-1}}\circ D_{\varepsilon_{j-2},\varepsilon_{j-1}}\circ \cdots \circ h_{\varepsilon_2}\circ D_{\varepsilon_1,\varepsilon_2}\circ i_{\varepsilon}.
\]
The component of the map $I$ sending coordinate $\varepsilon$ to coordinate $\varepsilon'$ is given via the formula
\[
I_{\varepsilon,\varepsilon'}:=\sum_{\varepsilon=\varepsilon_1<\cdots<\varepsilon_j=\varepsilon'}  h_{\varepsilon_{j}}\circ D_{\varepsilon_{j-1},\varepsilon_{j}}\circ \cdots \circ h_{\varepsilon_2}\circ D_{\varepsilon_1,\varepsilon_2}\circ i_{\varepsilon}.
\]
Similarly, $\betaPi$ is given by the formula
\[
\betaPi_{\varepsilon,\varepsilon'}:=\sum_{\varepsilon=\varepsilon_1<\cdots<\varepsilon_j=\varepsilon'}  \pi_{\varepsilon'}\circ D_{\varepsilon_{j-1},\varepsilon_j}\circ h_{\varepsilon_{j-1}}\circ D_{\varepsilon_{j-2},\varepsilon_{j-1}}\circ \cdots\circ D_{\varepsilon_1,\varepsilon_2} \circ h_{\varepsilon_1}.
\]

\subsection{Completion of the proof of Theorem~\mathrm{re}f{thm:main}}

We now describe the proof of Theorem~\mathrm{re}f{thm:main}. Firstly, there is no loss of generality in assuming that $G$ is connected, since if $G=G_1\sqcup G_2$ then $Y(G)\cong Y(G_1)\# Y(G_2)$, and both theories are tensorial under connected sum of 3-manifolds.

We consider forests of trees $G$, such that each component has a distinguished vertex, which we label as the root. We consider the following operations on rooted trees, from which any rooted tree may be obtained:
\betaegin{enumerate}[label= ($G$-\mathrm{a.r.}abic*), ref=$G$-\mathrm{a.r.}abic*]
\item\label{graph-1} Adding a valence 0 vertex (viewed as the root of its component).
\item\label{graph-2}  Joining two components together at their roots.
\item\label{graph-3} Adding a valence 1 vertex at the root of a component, and making the new vertex the new root.
\end{enumerate}

 We now extend the description of lattice homology in Proposition~\mathrm{re}f{prop:OSS-lattice} to a statement about bordered link surgery modules and the above operations on graphs. We form a chain complex $\widetilde{\mathcal{C}}_{\Lambda}(G)$ as follows. We pick a vertex $v_0\in V(G)$ which we label as the root vertex. We define the chain complex $\widetilde{\mathcal{C}}_{\Lambda}(G)$ by iteratively tensoring the bordered modules and bimodules as follows, in parallel with the above topological moves:
\betaegin{enumerate}[label= ($M$-\mathrm{a.r.}abic*), ref=$M$-\mathrm{a.r.}abic*]
\item\label{module-1}
We begin with a type-$D$ module for a solid tori $\mathcal{D}_{0}^{\mathcal{K}}$ for each valence 1 vertex of $G$ (other than $v_0$, if it has valence 1).
\item\label{module-2} We use the merge module ${}_{\mathcal{K}| \mathcal{K}} M^{\mathcal{K}}$ to tensor the type-$D$ modules for two rooted trees to form the type-$D$ module for the tree obtained by joining the two components together at their roots.
\item\label{module-3} We tensor with the type-$DA$ module ${}_{\mathcal{K}} \overline{\mathcal{H}}_{(w(v),0)}^{\mathcal{K}}$ of the Hopf link to add a valence 1 vertex at the root of a tree. Here $w(v)$ denotes the weight of the vertex $v$.
\item\label{module-4} We tensor the type-$AA$ bimodule ${}_{\mathcal{K}} [\mathcal{D}_{w(v_0)}]_{\mathbb{F}[U]}$ for the final root $v_0$.
\end{enumerate}

Note that since the bimodules are Alexander modules (see \cite{ZemBordered}*{Section~6}), the final type-$A$ action of $\mathbb{F}[U]$ extends to an action of $\mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}] $ on the completion of $\widetilde{\mathcal{C}}_{\Lambda}(G)$. 

We note that the complex $\widetilde{\mathcal{C}}_{\Lambda}(G)$ may be described as the algebraic complex obtained by using the connected sum formula for the hypercube maps in Theorem~\mathrm{re}f{thm:pairing-main}.  We prove the following:

\betaegin{prop}
\label{prop:tilde-C=lattice}
 The chain complex $\widetilde{\mathcal{C}}_{\Lambda}(G)$ is homotopy equivalent to $\mathbb{C}F(G)$, viewed as a type-$A$ module over $\mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}] $ with only $m_1$ and $m_2$ non-vanishing.
\end{prop}

\betaegin{rem}
One could also argue by considering the minimal models of the Hopf link ${}_{\mathcal{K}} \overline{\mathcal{Z}}_{(w(v),0)}^{\mathcal{K}}$ from \cite{ZemBordered}*{Section~17}, which have $\partialelta_j^1=0$ if $j\neq 2$. These are related to the dual knot formula of Hedden--Levine \cite{HeddenLevineSurgery} and Eftekhary \cite{EftekharyDuals}. We will give a more direct argument using the homological perturbation lemma.
\end{rem}

\betaegin{proof}
We note that the Hopf link Floer complex has the following filtration:
\[
\mathcal{C}FL(H)=\betaig(\betaegin{tikzcd} \Span_{\mathbb{F}[\mathscr{U}_1,\mathscr{V}_1,\mathscr{U}_2,\mathscr{V}_2]}(\ve{b},\ve{c})\mathrm{a.r.}[r, "\partial"]& \Span_{\mathbb{F}[\mathscr{U}_1,\mathscr{V}_1,\mathscr{U}_2,\mathscr{V}_2]}(\ve{a},\ve{d})
\end{tikzcd}\betaig).
\]
Here, $\ve{a}$, $\ve{b}$, $\ve{c}$ and $\ve{d}$ are the generators from Equation~\eqref{eq:Hopf-def}

First note that $\widetilde{\mathcal{C}}_{\Lambda}(G)$ has a filtration by $\mathbb{E}_\ell$ where $\ell=|V(G)|$, similar to $\mathbb{C}F(G)$ and $\mathcal{C}_{\Lambda}(L_G)$. That is, $\widetilde{\mathcal{C}}_{\Lambda}(G)$ is an $n$-dimensional hypercube of chain complexes.  Write $\widetilde{\mathcal{C}}_{\Lambda}(G)=(\widetilde{\mathcal{C}}_{\varepsilon}, \widetilde{D}_{\varepsilon,\varepsilon'})_{\varepsilon\in \mathbb{E}_n}$. 

For each $\varepsilon$, consider the subcomplex $\widetilde{\mathcal{C}}_{\varepsilon,\ve{s}}$ of $\widetilde{\mathcal{C}}_{\varepsilon}$ which lies in a single Alexander grading $\ve{s}\in \mathbb{H}(L_G)$.  The complexes $\mathcal{C}_{\varepsilon}$ are obtained by localizing a tensor product of  $(\ell-1)$-copies of the Hopf link complex $\mathcal{C}FL(H)$ at some of the $\mathscr{V}_i$ variables, and completing with respect to the $U_i$ and taking the direct product over Alexander gradings. In particular, $\widetilde{\mathcal{C}}_{\varepsilon,\ve{s}}$ has a filtration similar to the filtration on the Hopf link complex.  Write $\widetilde{c}_{\varepsilon,\ve{s}}$ for the chain complex $\widetilde{\mathcal{C}}_{\varepsilon,\ve{s}}$ before completing with respect to the $U_i$ variables. The complex $\widetilde{c}_{\varepsilon,\ve{s}}$ may be written as the following exact sequence:
\betaegin{equation}
\widetilde{c}_{\varepsilon,\ve{s}}=\betaegin{tikzcd} 0\mathrm{a.r.}[r]&\mathcal{F}^\ell_{\varepsilon,\ve{s}}\mathrm{a.r.}[r]& \cdots \mathrm{a.r.}[r]& \mathcal{F}^1_{\varepsilon,\ve{s}}.
\end{tikzcd}
\label{eq:long-exact-sequence}
\end{equation}
We refer to the superscript $i$ in $\mathcal{F}_{\varepsilon,\ve{s}}^i$ as the \emph{Hopf filtration} level. In Equation~\eqref{eq:long-exact-sequence}, $\mathcal{F}^i_{\varepsilon,\ve{s}}$ is generated over $\mathbb{F}[U_1,\partialots, U_{\ell}]$ by elementary tensors in $S_{\varepsilon}^{-1} \cdot \mathcal{C}FL(L_G)$ which contain $i-1$ factors which are $\ve{b}$ or $\ve{c}$, and $\ell-i$ factors which are  $\ve{a}$ or $\ve{d}$, and which have Alexander grading $\ve{s}$. Recall that $S_{\varepsilon}$ denotes the multiplicatively closed set generated by $\mathscr{V}_i$ where $\varepsilon_i=1$.

Index the link components so that the root vertex $v_0$ corresponds to the variable $U=U_1$. The homology group $H_*(\widetilde{c}_{\varepsilon,\ve{s}})$ is isomorphic to $\mathbb{F}[U_1]$ by \cite{OSPlumbed}*{Lemma~2.6} (see also \cite{OSSLattice}*{Lemma~4.2}). Furthermore, the generator may be taken to be an elementary tensor which has only factors of $\ve{a}$ and $\ve{d}$, as well as some of the $\mathscr{U}_i$ and $\mathscr{V}_i$ variables. In particular, the generator is supported in $\mathcal{F}_{\varepsilon,\ve{s}}^1$. It follows that the chain complex $\widetilde{c}_{\varepsilon,\ve{s}}$ is a free resolution of its homology over $\mathbb{F}[U_1,\partialots, U_\ell]$.

 Since $H_*(\widetilde{c}_{\varepsilon,\ve{s}})\cong \mathbb{F}[U_1]$ is a projective module over $\mathbb{F}[U_1]$, it is a basic exercise in homological algebra to show that the above exact sequence may be split over $\mathbb{F}[U_1]$. That is, we may decompose each $\mathcal{F}_{\varepsilon,\ve{s}}^{i}$ over $\mathbb{F}[U_1]$ into a direct sum of $\mathbb{F}[U_1]$-modules
 \[
 \mathcal{F}_{\varepsilon,\ve{s}}^i\cong \mathcal{F}_{\varepsilon,\ve{s}}^{i,l}\oplus \mathcal{F}_{\varepsilon,\ve{s}}^{i,r}
 \]
 so that $\partial$ maps each $\mathcal{F}^{i,r}_{\varepsilon,\ve{s}}$ isomorphically onto $\mathcal{F}^{i-1,l}_{\varepsilon,\ve{s}}$, for $1<i<\ell$, and $\partial$ vanishes on $\mathcal{F}^{i,r}_{\varepsilon,\ve{s}}$. Further, $\mathcal{F}^{\ell,l}_{\varepsilon,\ve{s}}=0$ and $\mathcal{F}_{\varepsilon,\ve{s}}^{1,r}$ projects isomorphically into $H_*(c_{\varepsilon,\ve{s}})$. We may then define maps $\pi_{\varepsilon,\ve{s}}$, $h_{\varepsilon,\ve{s}}^{i}$ and $i_{\varepsilon,\ve{s}}$ as in the following diagram
 \[
 \betaegin{tikzcd}[column sep=.8cm]
  \mathcal{F}^{\ell,r}_{\varepsilon,\ve{s}}
  	\mathrm{a.r.}[r, "\partial",swap]
  &
  \mathcal{F}^{\ell-1,l}_{\varepsilon,\ve{s}}\oplus  \mathcal{F}^{\ell-1,r}_{\varepsilon,\ve{s}}
  \mathrm{a.r.}[r, "\partial",swap]
  \mathrm{a.r.}[l, "h^{\ell-1}_{\varepsilon}",bend right,swap]
  &
  \cdots
  \mathrm{a.r.}[l,bend right, swap,"h^{\ell-2}_{\varepsilon,\ve{s}}"]
  \mathrm{a.r.}[r,swap, "\partial"]
  &
  \mathcal{F}^{2,l}_{\varepsilon,\ve{s}}
  \oplus
    \mathcal{F}^{2,r}_{\varepsilon,\ve{s}}
 \mathrm{a.r.}[r, "\partial",swap]
 \mathrm{a.r.}[l,bend right,swap, "h^2_{\varepsilon,\ve{s}}"]
 &
 {\mathcal{F}}^{1,l}_{\varepsilon,\ve{s}}
 \oplus
 {\mathcal{F}}^{1,r}_{\varepsilon,\ve{s}}
  \mathrm{a.r.}[l,bend right,swap, "h^1_{\varepsilon,\ve{s}}"]
  \mathrm{a.r.}[r,swap, "\pi_{\varepsilon,\ve{s}}"]
 &H_*(\widetilde{c}_{\varepsilon,\ve{s}})
 	\mathrm{a.r.}[l, bend right, swap, "i_{\varepsilon,\ve{s}}"]
 \end{tikzcd}
 \]
The map $\pi_{\varepsilon,\ve{s}}$ is the canonical projection map, and $i_{\varepsilon,\ve{s}}$ is a section of the map $\pi_{\varepsilon,\ve{s}}$, which we assume is compatible with the splitting described earlier. Similarly, $h_{\varepsilon,\ve{s}}^i$ is the inverse of $\partial|_{\mathcal{F}_{\varepsilon,\ve{s}}^{i,l}}$. Note that $i_{\varepsilon,\ve{s}}$ and $h_{\varepsilon,\ve{s}}^i$ are not generally $\mathbb{F}[U_1,\partialots, U_\ell]$-equivariant, and instead only $\mathbb{F}[U_1]$-equivariant.

We may define $\mathbb{F}[\hspace{-.5mm}[ U_1]\hspace{-.5mm}] $-equivariant maps
\[
\pi_{\varepsilon}\colon \widetilde{\mathcal{C}}_{\varepsilon}\to H_*(\widetilde{\mathcal{C}}_{\varepsilon}),\quad  i_{\varepsilon}\colon H_*(\widetilde{\mathcal{C}}_{\varepsilon})\to \widetilde{\mathcal{C}}_{\varepsilon}\quad \text{and} \quad  h_{\varepsilon}\colon \widetilde{\mathcal{C}}_{\varepsilon}\to \widetilde{\mathcal{C}}_{\varepsilon}
\]
which give a homotopy equivalence over $\mathbb{F}[\hspace{-.5mm}[ U_1]\hspace{-.5mm}] $ between $\widetilde{\mathcal{C}}_{\varepsilon}$ and $H_*(\widetilde{\mathcal{C}}_{\varepsilon})$, by taking the direct product of the maps $i_{\varepsilon,\ve{s}}$, $\pi_{\varepsilon,\ve{s}}$ and $h_{\varepsilon,\ve{s}}=\sum_{i=1}^\ell h^i_{\varepsilon,\ve{s}}$, and then completing with respect to the variables $U_1,\partialots, U_\ell$. To see that the maps $i_{\varepsilon,\ve{s}}$, $\pi_{\varepsilon,\ve{s}}$ and $h_{\varepsilon,\ve{s}}$ induce well-defined maps after completing with respect to $U_1,\partialots, U_\ell$, we argue as follows. The completion over $U_1,\partialots,U_\ell$ may be viewed as the completion with respect to the $I$-adic topology on $\mathbb{F}[U_1,\partialots, U_\ell]$, where $I$ is the ideal $(U_1,\partialots, U_\ell)$. See \cite{AtiyahMacdonald}*{Chapter~10}. Equivalently, since there are only finitely many generators over $\mathbb{F}[U_1,\partialots, U_\ell]$ and each $U_i$ has Maslov grading $-2$, we may describe the completion as having a fundamental system of open sets given by $M_i=\Span\{x: \gammar(x)\le i\}$ ranging over $i<0$.  The maps $i_{\varepsilon,\ve{s}}$ and $h_{\varepsilon,\ve{s}}$ are clearly continuous with respect to this topology, since they are homogeneously graded, and hence induce maps on the completion.

Applying the homological perturbation lemma for hypercubes, Lemma~\mathrm{re}f{lem:homological-perturbation-cubes}, we may transport the hypercube maps of $(\widetilde{\mathcal{C}}_{\varepsilon}, \widetilde{D}_{\varepsilon,\varepsilon'})$ to a hypercube with underlying groups $H_*(\widetilde{\mathcal{C}}_{\varepsilon})$. This coincides with the underlying group of the lattice complex by Proposition~\mathrm{re}f{prop:OSS-lattice}. To see that the resulting hypercube is the lattice complex, it is sufficient to show that there are no higher length hypercube maps when we apply the homological perturbation lemma. This is seen directly, as follows. The recipe from the homological perturbation lemma for hypercubes is to include via $i_{\varepsilon}$, then sequentially compose cube maps and the homotopies $h_{\varepsilon}$, and then finally apply $\pi_{\varepsilon'}$. Note that $h_{\varepsilon}$ strictly increases the Hopf filtration level in the sequence in Equation~\eqref{eq:long-exact-sequence}, and $\pi_{\varepsilon}$ is non-vanishing on only the lowest Hopf filtration level. Additionally, using Proposition~\mathrm{re}f{prop:Hopf-link-large-model} and the tensor product formula in Theorem~\mathrm{re}f{thm:pairing-main}, we see that the length 1 maps of $\widetilde{\mathcal{C}}_{\Lambda}(G)$ preserve the Hopf filtration level, while the higher length maps strictly increase the Hopf filtration level. In particular, the only way for a summand from Lemma~\mathrm{re}f{lem:homological-perturbation-cubes} to contribute is for there to be no $h_{\varepsilon}$ factors, and for the hypercube arrow to be length 1. Hence, the transported hypercube maps are exactly the maps induced on homology by the length one maps of $\widetilde{\mathcal{C}}_{\Lambda}(G)$. This is exactly lattice homology by Proposition~\mathrm{re}f{prop:OSS-lattice}. The proof is complete.
\end{proof}

\betaegin{rem} The above argument fails at the last step if we use the $\mathcal{H}$-models instead of the $\overline \mathcal{H}$ models. This is because the length 2 map of $\mathcal{H}$ has a term which decreases the Hopf filtration level, so could potentially contribute to higher length terms in the hypercube structure maps obtained from the homological perturbation lemma.
\end{rem}

\betaegin{lem}
\label{lem:rational-homology-plumbings}
Suppose that $Y(G)$ is a 3-manifold obtained by plumbing along a tree $G$ and that $b_1(Y(G))=0$. Let $v$ be a vertex of $G$. We may view $Y(G)$ as the surgery on a knot $K_1\# \cdots\# K_n\subseteq Y_1\#\cdots \# Y_n$ where  $(Y_1,K_1),\partialots, (Y_n,K_n)$ are the 3-manifold-knot pairs corresponding to the component of $G$ obtained by splitting $v$ into $n$ valence-1 vertices. Then $b_1(Y_1)+\cdots+b_1(Y_n)\le 1$. In particular, at most one $K_i$ is homologically essential in $Y_i$.
\end{lem}
\betaegin{proof}
We have $b_1(Y_1\# \cdots \# Y_n)=b_1(Y_1)+\cdots+b_1(Y_n)$. On the other hand, $Y(G)$ is obtained by performing Dehn surgery $K_1\#\cdots \# K_n\subseteq Y_1\#\cdots\# Y_n$ and $b_1(Y(G))=0$.  Surgery on a knot can reduce $b_1$ by at most one, so the claim follows.
\end{proof}

We now complete the proof of Theorem~\mathrm{re}f{thm:main} when $b_1(Y(G))=0$:

\betaegin{proof}[Proof of Theorem~\mathrm{re}f{thm:main} when $b_1(Y(G))=0$]
To compute $\ve{\mathbb{C}F}^-(Y(G))\cong \mathcal{C}_{\Lambda}(L_G)$, we can tensor bordered bimodules similar to ~\eqref{module-1}--\eqref{module-4}, as in our construction of $\widetilde{\mathcal{C}}_{\Lambda}(G)$. The only difference is that instead of the merge module $ M$, we must use the pair-of-pants modules $W_r$ or $W_l$. Note that we may construct $Y(G)$ by iteratively performing topological moves parallel to the algebraic modules~\eqref{module-1}--\eqref{module-4}. By Lemma~\mathrm{re}f{lem:rational-homology-plumbings}, when constructing plumbed 3-manifolds which are rational homology 3-spheres by iteratively taking connected sums and taking duals, we will never take the connected sum of two knots which are both homologically essential. Hence, we may apply Corollary~\mathrm{re}f{cor:simplify-tensor-product} to replace the pair-of-pants modules with the merge module, and the homotopy type of the resulting type-$D$ module over $\mathcal{K}$ will be unchanged. In particular, we see that for some choice of arc system $\mathscr{A}$ on $L_G$, we have
\[
\mathcal{C}_{\Lambda}(L_G,\mathscr{A})_{\mathbb{F}[U]}\simeq \widetilde{\mathcal{C}}_{\Lambda}(L_G)_{\mathbb{F}[U]},
\]
so the main result follows from Proposition~\mathrm{re}f{prop:tilde-C=lattice}.

We now address the claim about the relative grading. We recall from \cite{MOIntegerSurgery}*{Section~9.3} that when $b_1(Y(G))=0$, the link surgery hypercube possesses a uniquely specified relative $\mathbb{Z}$-grading on each $\Spin^c$ structure. Furthermore, this relative grading clearly coincides with the relative grading on the lattice complex. Note that in our proof, we showed that the link surgery complex $\mathcal{C}_{\Lambda}(L_G)$ coincided with the tensor product of the bordered modules in~\eqref{module-1}--\eqref{module-4} only up to homotopy equivalence. However in the case that $b_1(Y(G))=0$, the maps appearing in this homotopy equivalence only involve maps which already appear in the differential of the link surgery formula, or involve projections onto complexes in different $\varepsilon$ and $\ve{s}\in \mathbb{H}(L)$. In particular, the homotopy equivalence will be grading preserving.
\end{proof}

We now consider the case that $b_1(Y(G))>0$:
\betaegin{proof}[Proof of Theorem~\mathrm{re}f{thm:main} when $b_1(Y(G))>0$]
The proof is much the same as when $b_1(Y(G))=0$. The main difference is relating tensor products with the pair-of-pants bimodules with tensor products using the merge module. We can no longer use Corollary~\mathrm{re}f{cor:simplify-tensor-product} to eliminate the terms involving the homology action. Instead, we use Propositions~\mathrm{re}f{prop:merge=pair-of-pants} and ~\mathrm{re}f{prop:finite-generation}.
\end{proof}

\betaegin{rem} When $b_1(Y(G))>0$, the homotopy equivalence relating the tensor products obtained using the merge modules and the pair-of-pants bimodules is obtained using Corollary~\mathrm{re}f{cor:eliminate-homology-action}, which is fundamentally based on the classification theorem for finitely generated chain complexes over a PID. In particular, it is not as clear from this perspective how the homotopy equivalence interacts with the relative grading on the link surgery formula. We will pursue this question in a future work by considering group valued gradings on the bordered link surgery modules, in the spirit of \cite{LOTBordered}*{Section~3.3}.
\end{rem}

\section{Plumbed 3-manifolds with $b_1>0$}
\label{sec:plumbed-H1-action}
In this section, we state a refinement of N\'{e}methi's conjecture for plumbed manifolds with $b_1>0$. In analogy to the action of $H_1(S^3_\Lambda(L))/\mathbb{T}ors$ on the link surgery complex from Section~\mathrm{re}f{sec:algebraic-action-link-surgery}, we now describe an action of $H_1(Y(G))/\mathbb{T}ors$ on lattice homology. If $\mu_i$ denotes the meridian of the component $K_i\subseteq L_G$, we define
\[
\mathfrak{A}_{[\mu_i]}([K,E])=U^{a_{v_i}[K,E]} [K,E-v_i],
\]
extended equivariantly over $U$. (This is a term in the differential $\partial$).
If $\gamma=a_1[\mu_1]+\cdots+ a_\ell [\mu_\ell]\in H_1(Y(G))/\mathbb{T}ors$, we define
\[
\mathfrak{A}_{[\gamma]}:=a_1\mathfrak{A}_{[\mu_1]}+\cdots+ a_\ell \mathfrak{A}_{[\mu_\ell]}.
\]
\betaegin{lem} The above action of $\mathbb{Z}^{\ell}$ on $\mathbb{C}F(G)$ descends to a well-defined action of $H_1(Y(G))/\mathbb{T}ors$ up chain  homotopy.
\end{lem}
The proof is essentially identical to the proof of Lemma~\mathrm{re}f{lem:algebraic-homology-action-link-surgery} for the link surgery formula. One replaces the lattice $\mathbb{H}(L)$ with $\mathbb{C}har(X(G))$ and the proof proceeds by an easy translation.

\betaegin{conj} $\ve{\mathit{HF}}^-(Y(G))$ is isomorphic to $\mathbb{HF}(G)$ as a module over $\mathbb{F}[\hspace{-.5mm}[ U]\hspace{-.5mm}] \otimes \Lambda^* H_1(Y(G))/\mathbb{T}ors$.
\end{conj}

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