  Reducing a 6d fivebrane theory on a 3-manifold $Y$ gives a $q$-series
3-manifold invariant $\widehat{Z}(Y)$. We analyse the large-$N$ behaviour of
$F_K=\widehat{Z}(M_K)$, where $M_K$ is the complement of a knot $K$ in the
3-sphere, and explore the relationship between an $a$-deformed ($a=q^N$)
version of $F_{K}$ and HOMFLY-PT polynomials. On the one hand, in combination
with counts of holomorphic annuli on knot complements, this gives an
enumerative interpretation of $F_K$ in terms of counts of open holomorphic
curves. On the other, it leads to closed form expressions for $a$-deformed
$F_K$ for $(2,2p+1)$-torus knots. They suggest a further $t$-deformation based
on superpolynomials, which can be used to obtain a $t$-deformation of ADO
polynomials, expected to be related to categorification. Moreover, studying how
$F_K$ transforms under natural geometric operations on $K$ indicates relations
to quantum modularity in a new setting.

High Energy Physics - Theory;Geometric Topology;Number Theory;Quantum Algebra;Symplectic Geometry