text:
97% Of Your Genome Is Junk DNA
Your DNA is made up of about 3.2 billion bases. Around 97 percent of that doesn't do anything—at least, we don't think it does.
What This Means
When scientists first began sequencing DNA in the 1970s, they noticed something strange: along the familiar double-helix were small pieces of DNA that coded for particular proteins, followed by long strings that appeared to do nothing at all. The human genome, in other words, was mostly fluff.
But like the bolts left over after assembling an IKEA dresser, it turns out that much of this so-called junk may serve a purpose. Over time, some scientists started to suspect that certain strings of DNA had once been junk, but evolved the ability to code for proteins. And in 2012, a research program called ENCODE found that around 75% of junk DNA undergoes transcription (the first step to a gene actually being expressed). However, not all scientists agree that this removes their "junk" status, since that doesn't necessarily mean that they have any biological function.
Smart Graphic
Why It's Important
The fact that our genome isn't an orderly chain of useful genes, but instead a heap of useful, halfway useful, and totally useless strings of DNA is yet another bit of evidence for the fact that we evolved, and weren't designed. Evolution works blindly through random mutations that sometimes kill off a species and sometimes help it survive, and as a result it leaves behind useless mutations that just stay put over the millennia. To call so much of your DNA junk might sound like a bad thing, but think of it this way: it's just a genetic keepsake from generations of evolution.
Editors' Picks: Our Favorite Videos About Genes
98% Of Your DNA Is Junk
Your genome is a big ol' mess.
Genes Explained
What are you made of, really?
What Everything In Your Body Is Really Made Of? Genes Explained.
– TestTube Plus
Are Genes Really Selfish?
If genes only care about their survival, you'd think it would make all organisms greedy. In that case, how did altruism evolve?
Are Genes Really Selfish? - Instant Egghead #57
– Scientific American
Written by Ashley Hamer December 19, 2016
A Baby Was Born With The DNA Of 3 People
With Epigenetics, Your DNA Can Change Within Your Lifetime


text:
Ranking the 6 Paul George, LeBron James Recruiting Billboards in LA
The 5 Best Commercials Featuring the NBA's Top 2018 Free Agents
Brett Regan, June 29, 2018 2:26 pm
Which Cities Should the NBA Expand To?
Joe Grobeck, January 20, 2021 12:42 pm
Inside Rick Fox's Staggering Hollywood Love Life
Joe Grobeck, January 19, 2021 10:45 am
Swin Cash's NBA Front Office Job is Perfect for Retired Stars
LeBron James will become an unrestricted free agent. The King might place his throne somewhere else. The best player on the planet can go anywhere he so chooses and shake up the entire NBA once again. Let the madness officially begin.
LeBron James' agent informed the Cavs he will not exercise his $35.6 million option and thus will become an unrestricted free agent, sources told @clevelanddotcom … Story coming
— Joe Vardon (@joevardon) June 29, 2018
NBA Free Agency is serious business, but that doesn't mean it can't be fun. Despite the relentless rumors, which will make your head spin, there are billboards and fake trade scenarios that make this time of year highly entertaining.
This summer is a little different, though. There are usually big-time players on the market each and every year, but 2018 features some of the game's biggest stars, including the absolute biggest in James.
So in light of this not-so-shocking news of James opting out of his contract with the Cleveland Cavaliers and the craziness that will certainly ensue, let's take a look at the top commercials from the best NBA free agents.
1. LeBron James
There are really too many good options here. Seriously, it impossible to choose. From "The Rise" to the "Tattoo," James has dominated this decade with commercials like no other NBA star. But since it is the summer, let's throw it back to an all-time classic.
Oh, and don't forget this just happened, too.
Everyone asking me if I really jumped earlier from my story early on today, well here you go! Haters will say it's fake(videoshopped). 🤣🤣🤷🏾♂. P.S excuse my language but it's Still #BlackExcellence✊🏾 #JamesGang👑
A post shared by LeBron James (@kingjames) on Jun 28, 2018 at 8:18pm PDT
2. Kevin Durant
Before he was a scoring champion with the Oklahoma City Thunder, NBA MVP, and two-time champion with the Golden State Warriors, Durant was this skinny, up-and-coming rapper named Velvet Hoop. Well, maybe he wasn't planning a rap career, but we can still appreciate the simpler times of the young, buck-getting Durant before he re-signs with the Warriors.
3. Paul George
Paul George doesn't really have quite the number of spots James and Durant have, and understandably so, yet this one is pure gold. Forget this is likely going to be two teams ago, George literally shot a rainbow, walked back to the bench with so much swagger, and had time to sip on his Gatorade before it went in. Legendary stuff. Let's get more of this in real life, whether he goes to the Los Angeles Lakers or not.
4. Chris Paul
The catalog of State Farm commercials featuring Paul is quite solid from top to bottom. However, "The Hoopers" took it to the next level. In this scene, Paul is competitive and snarky, yet hilarious. This is how he plays, this is peak Paul, and this how he needs to be heading into next season when he likely re-signs with the Houston Rockets.
5. DeMarcus Cousins
https://www.youtube.com/watch?v=DlAYur_YAaU
This is perfect on so many levels. Boogie is as misunderstood as he is talented, and he is easily one of the best big men in the entire NBA. Where Cousins might landed following his Achilles injury with the New Orleans Pelicans is probably more up in the air than any player on this list, but it's safe to say that no matter where he goes, the drama and reputation, rightly or wrongly, will follow.
Read More: Billboards to Recruit Paul George, LeBron James Take LA by Storm
About the author: Brett Regan, Editor
With over 10 years of sports writing experience, Brett has covered some of the top local, regional, and national sporting events in the Heartland for both print and digital platforms. He is a graduate of Kansas State University and resides in Austin, Texas.

redpajama_set_name: RedPajamaCommonCrawl


The Curacao Tourist Board says the continuous support of the CTB and partners to promote the Curacao brand in North America is reaping rewards for the island.
Increase in Canadians, new cruise port and hotels for the D.R.
The Dominican Republic has once again solidified its reputation for being a top vacation destination, welcoming over 900,000 Canadians in 2018.
Virtuoso's Canadian footprint has expanded to more than 2,200 advisors, up 18% since 2017, with year-over-year sales growth pegged at 26% for a total of US$2.1 billion.
The Bahamas has achieved its highest number of international arrivals ever, thanks in part to a boost in lift from source markets including Canada. The strong numbers have resulted in record hotel occupancy levels.


What the hell @ECISVEEP is doing?
This video is doing the rounds, where it is said that #Gurgaon #BJP MLA faced angry residents, & had to slip away without addressing the planned meeting. I guess, people now want the political parties to talk on core issues.
As per a voter, Dhara Singh, 6 persons observed that irrespective of whichever button was being pressed, vote was going to BJP. Booth-in-charge too agreed there was a problem. Voters are now demanding verification of their votes.
As per a voter, Dhara Singh, 6 persons observed that irrespective of whichever button was being pressed, vote was going to BJP. Booth-in-charge too agreed there was a problem.


Do you have a product that you would like to sell through Wallace & Carey? Why not send us a message?? One of our teammates will get in touch with you as soon as possible to determine if your product is a great match for our customer base!!
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pply a generous amount of Moroccanoil® Hydrating Mask Light to towel-dried hair and comb through. Leave on for 5-7 minutes and rinse thoroughly. No heat required. Use 1-2 times weekly.
TIP: For deeper conditioning, mix a few drops of Moroccanoil Treatment in with this hair mask.


text:
Barbara Almond (June 6, 1938 – March 6, 2016) was an American psychiatrist and psychoanalyst. She authored books on psychiatry, including The Monster Within: The Hidden Side of Motherhood (2010).

Biography
Almond was born Barbara Mary Rosenthal in The Bronx. Her father was an actuary and her mother was a teacher. She attended The High School of Music & Art in Manhattan before graduating from Antioch College and Yale University Medical School. Almond had a private practice in Palo Alto, California, and taught at Stanford University and the San Francisco Center for Psychoanalysis.

With her husband Richard, also a psychiatrist, Almond wrote The Therapeutic Narrative, a book about psychiatric conditions in literary characters. In 2010, she authored The Monster Within: The Hidden Side of Motherhood, which discussed the feelings of ambivalence that women often held toward motherhood. In a review in The Washington Post, Carolyn See wrote that Almond "calms the reader, suggesting that we can only do our best and trust that our ambivalence is more than compensated for by our devotion and love."

Almond and her husband were married for 53 years. In her leisure time, Almond played piano. She was diagnosed with bladder cancer in early 2015 and continued practicing psychiatry until two months before her death.

References

1938 births
2016 deaths
Writers from the Bronx
American psychiatrists
Deaths from bladder cancer
Antioch College alumni
Yale School of Medicine alumni
Stanford University School of Medicine faculty
American women psychiatrists
The High School of Music & Art alumni
Scientists from the Bronx
Deaths from cancer in California
21st-century American women

redpajama_set_name: RedPajamaWikipedia


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Jeffrey J. Wetzel | Dickie, McCamey & Chilcote, P.C.
Jeffrey J. Wetzel is a shareholder of Dickie, McCamey & Chilcote, P.C.; Co-Chair of the Health Law Business Group; and member of the firm's Employment Committee. He concentrates his litigation practice in the areas of medical malpractice defense, employment law, general liability, and appellate practice. In his 14 years of practice, Mr. Wetzel has represented various hospitals and health care providers, including some of the nation's largest health care institutions and dialysis clinics, as well as community hospitals, physician practice groups, and individual nurses and physicians. He regularly advises clients on issues including regulatory compliance, litigation avoidance, and employment-related matters.
Mr. Wetzel is a NITA (National Institute for Trial Advocacy) trained trial attorney.
Mr. Wetzel has lectured on issues of appellate practice, medical malpractice defense, and the intersection of psychiatry and the law.


Why intervention (with the use of a mirror) is different from person to person may help to illuminate the efficacy of this as a part of treatment. Furthermore, we may also more thoroughly understand the mechanism of cure for a patient.... The client can use a journal, especially if the therapy is done at home, as a tool to help with follow-up and to inform the therapists of the progress of the therapy (mirror therapy schedule, client's experiences, exercises done, etc.).
Some have suggested that "Man in the Mirror" takes its inspiration from a poem known as "Guy in the Mirror" or "Guy in the Glass." The poem, written in 1934 by Dale Wimbrow, became widely popular a...... Mirroring, a technique used in sales, can work well when done properly or create a disastrous situation when used incorrectly. Basically, one person adopts the physical and verbal behaviors of another as a way to build rapport and agreement during the selling process.
MIRROR WORK: 21 DAYS TO HEAL YOUR LIFE, is the first book dedicated to my signature practice A positive affirmation opens the door. It's a beginning point on the path to change.
Why intervention (with the use of a mirror) is different from person to person may help to illuminate the efficacy of this as a part of treatment. Furthermore, we may also more thoroughly understand the mechanism of cure for a patient.


Gigler? Where the hell are you?
I've emailed - no response.
I've done a support ticket - no response.
I've tweeted - twice. No response.
I've face booked. - twice. No response.
I've called half a dozen times. Just asked to leave a message. I have.
Zero responses on all fronts.
Utter utter shite customer service.
I've had no internet for 24 hours. No responses or even a posting of why.


text:
\section{Introduction}\scalefont{1.05}
\par{The topological string theory amplitudes are useful in the study of the BPS sector of the string compactifications. ${\cal N}=2$ topological strings on Calabi-Yau threefolds capture the quantum numbers of the BPS states obtained by wrapping branes on holomorphic curves \cite{GV}. When the Calabi-Yau threefold is non-compact, the topological string amplitudes  completely determine the prepotential, and the higher genus corrections to it, of the gauge theory which lives in the spacetime \cite{Nekrasov}. Surprisingly these amplitudes also determine the superconformal index of the 5D superconformal theories which appear by M-theory compactification on non-compact Calabi-Yau threefolds \cite{Vafa:2012fi}. }

\par{Advances in the instanton computation of ${\cal N}=2$ gauge theories motivated the refinement of the topological string amplitudes for non-compact Calabi-Yau threefolds \cite{Moore:1997dj,Nekrasov,HIV} . From the target space interpretation of the topological string theory the refinement corresponds to an explicit determination of the quantum numbers of BPS particles in M-theory compactifications on the same Calabi-Yau threefold \cite{GV,HIV} with respect to the $SO(4)=SU(2)_{L}\times SU(2)_{R}$ part of the 5D Lorentz group. Therefore it is particularly useful to be able to calculate these refined amplitudes for a non-compact Calabi-Yau threefold. The topological vertex completely solved the problem of determining the usual topological string amplitudes for non-compact toric Calabi-Yau threefolds \cite{AKMV}. The refined topological vertex formulation extends the usual one and calculates the refined amplitudes for a certain class of non-compact toric Calabi-Yau threefolds \cite{IKV}.  The class of Calabi-Yau threefolds for which the refined vertex formalism works includes the geometries giving rise to gauge theory in spactime when type IIA strings are compactified on these Calabi-Yau threefolds. However, there exist toric Calabi-Yau threefolds which do not give rise to a gauge theory, such as the local ${\bf P}^{2}$, and the refined topological vertex formalism can not be applied to such geometries directly\footnote{However, BPS multiplicities can be calculated by appropriately identifying the curves that survive the blow down from a larger geometry, to which refined vertex formalism can be applied, in which it can be embedded. For local ${\bf P}^2$ the BPS multiplicities were calculated in \cite{IKV}}.}

\par{In this note, we introduce a complementary topological vertex, $T_{\lambda\,\mu\,\nu}(t,q)$, which together with the refined topological vertex extends the formalism to all non-compact toric Calabi-Yau threefolds. We believe that this new vertex is closely related to the ``index vertex" of Okounkov and Nekrasov \cite{ONtalk}. }

\par{The paper is organized as follows: in section 2, we discuss the origin of the refined topological vertex, its relation with instanton calculus of 4D gauge theory and the appearance of a preferred direction. In section 3, we discuss the case of local $\mathbf{P}^2$ and show the need for a new vertex. In this section we also calculate the refined partition function of the local $\mathbf{P}^2$. In section 4, we calculate the new vertex. Section 5 is reserved to the gluing rules of the new topological vertex with the refined topological vertex. We argue how the new vertex together with the refined vertex completely solve the problem of calculating the refined amplitudes for toric Calabi-Yau threefolds. In section 6, we give some applications of the new vertex.  In section 7, we discuss the relation of our approach to the refined Chern-Simons theory. Some technical points are included in appendix A. Notation and identities used are given in appendix B. }

\section{The refined vertex and gauge theory}

The geometric engineering provides a beautiful link connecting the spacetime physics with the geometry of the Calabi-Yau threefold. Type IIA strings compactified on Calabi-Yau threefolds give rise to ${\cal N}=2$ theory on the spacetime. The prepotential of the ${\cal N}=2$ theory on the spacetime is given by the genus zero amplitude of the topological strings propagating on the Calabi-Yau threefold. The higher genus amplitudes of the Calabi-Yau threefold appear when the gauge theory is coupled with gravity and, in case spacetime theory is a gauge theory coming from a toric Calabi-Yau threefold, can be calculated using the instanton calculus \cite{Nekrasov}. The K\"ahler parameters of the Calabi-Yau geometry are related, on the gauge theory side, to the expectation values of scalars in the vector multiplet $\vec{\bf a}$, the mass of the hypermultiplets and the gauge couplings. The string coupling constant is given by the equivariant parameter of the $U(1)$ action on $\mathbb{C}^2$:
\begin{eqnarray}\displaystyle
(z_{1},z_{2})\in \mathbb{C}^2\mapsto (q\,z_{1},q^{-1}\,z_{2})\,,\,\,\,\,q=e^{i\epsilon}\,.
\end{eqnarray}
This $U(1)$ action lifts to an action on the instanton moduli spaces with isolated fixed points labelled by the partitions of the instanton charge. For the $SU(N)$ gauge group the isolated fixed points of the $k$ instanton moduli space are labelled by $N$ partitions $\{Y_{1},Y_{2},\cdots, Y_{N}\}$ such that $\sum_{a}|Y_{a}|=k$, where $Y_{a}=\{Y_{a,1}\geq Y_{a,2}\geq Y_{a,3}\geq \cdots \,\,|Y_{a,i}\in \mathbb{Z}_{\geq 0}\}$ and $|Y_{a}|=\sum_{i}Y_{a,i}$. The instanton partition function is given by
\begin{eqnarray}\displaystyle
Z_{instanton}=\sum_{k=0}^{\infty}Q^{k}\,Z_{k}(\vec{\bf a},\epsilon)=\sum_{Y_{1},Y_{2},\cdots, Y_{N}}Q^{|Y_{1}|+Y_{2}|+\cdots +|Y_{N}|}\,Z_{Y_{1},Y_{2},\cdots, Y_{N}}(\vec{\bf a},\epsilon),
\end{eqnarray}
where $Z_{Y_{1},Y_{2},\cdots, Y_{N}}$ is the contribution from the fixed point labelled by $\{Y_{1},Y_{2},\cdots, Y_{N}\}$
\begin{eqnarray}\displaystyle
&&Z_{Y_{1},Y_{2},\cdots,Y_{N}}(\vec{\bf a},\epsilon)=\prod_{a,b=1}^{N}N_{Y_{a}\,,Y_{b}}({\bf a},\epsilon),\\\nonumber
&&N_{Y_{a}\,,Y_{b}}=\left[\prod_{(i,j)\in Y_{a}}\Big(1-e^{i(\alpha_{a}-\alpha_{b})}q^{Y^{t}_{b,j}+Y_{a,i}-i-j+1}\Big)\Big(1-e^{i(\alpha_{a}-\alpha_{b})}\,q^{-Y^{t}_{a,j}-Y_{b,i}+i+j-1}\Big)\right]^{-1}.
\end{eqnarray}
The full partition function also includes the perturbative part $Z_{pert}$\footnote{$\sum_{a=1}^{N}\alpha_{a}=0$}
\begin{eqnarray}\displaystyle
Z_{pert}=\left[\prod_{1\leq a<b\leq N}\Big(1-e^{i(\alpha_{a}-\alpha_{b})}q^{i}\,t^{j-1}\Big)\Big(1-e^{i(\alpha_{a}-\alpha_{b})}q^{i-1}\,t^{j}\Big)\right]^{-1}.
\end{eqnarray}

The geometric engineering relates the full partition function of the gauge theory
\begin{eqnarray}\displaystyle\nonumber
Z(Q,\vec{\bf a},\epsilon)=Z_{pert}(\vec{\bf a},\epsilon)\times Z_{instanton}(Q,\vec{\bf a},\epsilon)
\end{eqnarray}
with the partition function of topological strings propagating on a Calabi-Yau threefold \cite{Katz:1997eq}. In the case of the pure ${\cal N}=2\,\,\,SU(N)$ gauge theory we are discussing the Calabi-Yau threefold is the $A_{N-1}$ geometry fibered over a ${\mathbf P}^1$. The BPS particles in the theory are given by branes wrapping various cycles in the Calabi-Yau threefold \cite{KKV}. The $k$ instanton contribution to the gauge theory partition function comes from worldsheet instanton wrapping $k$ times the base ${\mathbf P}^1$ and arbitrary wrapping on the fiber ${\mathbf P}^1$'s of the $A_{N-1}$ fiber. The partition function encodes the BPS spin content of the particles coming from wrapping M2-branes on holomorphic curves. In the M-theory compactification on the Calabi-Yau threefold each BPS particle has a mass equal to the area of the curve and has $SO(4)=SU(2)_{L}\times SU(2)_{R}$ quantum numbers which we denote by $(j_{L},j_{R})$. The partition function is then given by \cite{GV}
\begin{eqnarray}\displaystyle
Z=\prod_{\beta\in H_{2}(X,\mathbb{Z})}\prod_{k_{L}=-j_{L}}^{+j_{L}}\prod_{m=1}^{\infty}\Big(1-Q_{\beta}\,q^{2k_{L}+m}\Big)^{(-1)^{2j_{L}+1}m\,N_{\beta}^{j_{L}}},
\end{eqnarray}
where $Q_{\beta}=e^{-\int_{\beta}\omega}$ ($\omega$ being the complexified K\"ahler class) and
\begin{eqnarray}\displaystyle
N_{\beta}^{j_{L}}=\sum_{j_{R}}(-1)^{2j_{R}}N_{\beta}^{j_{L},j_{R}}\,,
\end{eqnarray}
$N_{\beta}^{j_{L},j_{R}}$ being the number of particles with charge $\beta$ and spin $(j_{L},j_{R})$.

The refinement of topological strings was motivated by the fact that the equivariant localization computation mentioned above can also be carried out using the $U(1)\times U(1)$ action on $\mathbb{C}^2$
\begin{eqnarray}\displaystyle
(z_{1},z_{2})\mapsto (q\,z_{1},t^{-1}\,z_{2})\,,\,\,\,\,(q,t)=(e^{i\epsilon_{1}},e^{-i\epsilon_{2}}),
\end{eqnarray}
in which case the gauge theory partition function depends on $\epsilon_{1}$ and $\epsilon_{2}$. The fixed points of the instanton moduli space under this action are still as before. The only difference being that $N_{Y_{a},Y_{b}}$ now depend on both $q$ and $t$ \cite{Nekrasov, nakajima},
\begin{eqnarray}\displaystyle\nonumber
N_{Y_{a}\,,Y_{b}}=\left[\prod_{(i,j)\in Y_{a}}\Big(1-e^{i(\alpha_{a}-\alpha_{b})}t^{Y^{t}_{b,j}-i}\,q^{Y_{a,i}-j+1}\Big)\Big(1-e^{i(\alpha_{a}-\alpha_{b})}\,
t^{-Y^{t}_{a,j}+i-1}\,q^{-Y_{b,i}+j}\Big)\right]^{-1}\,.
\end{eqnarray}

Once the condition on the equivariant parameters is waived the general expansion defines the refinement of the topological string partition function. This is called refinement since it couples to the individual degenaracies with respect to the left-right spins \cite{HIV},
\begin{align}\label{sym}
Z_{refined}&=\prod_{\beta\in H_{2}(X,\mathbb{Z})}\prod_{k_{L}=-j_{L}}^{+j_{L}}\prod_{k_{R}=-j_{R}}^{+j_{R}}\prod_{m_{1},m_{2}=1}^{\infty}
\Big(1-Q_{\beta}\,t^{k_{L}-k_{R}+m_{1}-\frac{1}{2}}q^{k_{L}+k_{R}+m_{2}-\frac{1}{2}}\Big)^{M_{\beta}^{j_{L},j_{R}}},\,\\ \nonumber
&M_{\beta}^{j_{L},j_{R}}=(-1)^{2j_{L}+2j_{R}}\,N_{\beta}^{j_{L},j_{R}}.
\end{align}

The refinement has been studied both from the A-model side as well as the B-model point of view \cite{HIV,IKV,AS,HH,ACDKV,B1,B2,B3}. On the A-model side the refined topological vertex provides a powerful method for calculating the open and the closed partition functions for a certain class of toric Calabi-Yau threefolds \cite{IKV}. The geometries for which the refined vertex formalism can be used are essentially those which give rise to ${\cal N}=2$ gauge theories in four dimensions via geometric engineering. Some simple Calabi-Yau threefolds such as local ${\bf P}^2$ (total space of ${\cal O}(-3)$ bundle on ${\bf P}^2$) do not give rise to gauge theory upon type IIA compactification. It is not possible to directly use the refined topological vertex formalism to write down the partition function for this Calabi-Yau threefold. In \cite{IKV} a different method was employed to determine the quantum numbers of the BPS states coming from branes wrapping holomorphic curves in  local ${\bf P}^2$.

The refined topological vertex formalism solves the problem of finding the all genus partition function for the toric Calabi-Yau threefolds which engineer $U(N_{1})\times U(N_{2})\times \cdots \times U(N_{k})$ theories (including the quivers and matter multiplets in different representations) \cite{IKV}. The Calabi-Yau geometry is divided in ${\mathbb C}^{3}$ patches and the partition function is computed by suitably gluing the patches. The refined topological vertex is a function of two parameters $(q,t)\coloneqq(e^{i\epsilon_{1}},e^{-i\epsilon_{2}})$, labelled by three Young diagrams and it has the following form \cite{IKV} (see Appendix B for notation and conventions being used):
\begin{eqnarray}\displaystyle\nonumber
C_{\lambda\,\mu\,\nu}(t,q)&=&\Big(\frac{q}{t}\Big)^{\frac{\Arrowvert\mu\Arrowvert^2}{2}}\,t^{\frac{\kappa(\mu)}{2}}\,q^{\frac{\Arrowvert\nu\Arrowvert^2}{2}}\,\widetilde{Z}_{\nu}(t,q)
\sum_{\eta}\Big(\frac{q}{t}\Big)^{\frac{|\eta|+|\lambda|-|\mu|}{2}}\,s_{\lambda^{t}/\eta}(t^{-\rho}\,q^{-\nu})\,s_{\mu/\eta}(t^{-\nu^t}\,q^{-\rho}),\\
\end{eqnarray}
where $s_{\lambda/\eta}(\mathbf{x})$ is the skew-Schur function and $\widetilde{Z}_{\nu}(t,q)$ is given by
\begin{eqnarray}\displaystyle\nonumber
\widetilde{Z}_{\nu}(t,q)&=&\prod_{(i,j)\in \nu}\Big(1-q^{a(i,j)}\,t^{\ell(i,j)+1}\Big)^{-1}\,.
\end{eqnarray}
The function $C_{\lambda\,\mu\,\nu}(t,q)\times M(t,q)$, where $M(t,q)=\prod_{i,j=1}^{\infty}(1-q^{i}\,t^{j-1})^{-1}$ is a refinement of the MacMahon function, is a combinatorial object just like the usual topological vertex \cite{ORV}. $C_{\lambda\,\mu\,\nu}(t,q)\times M(t,q)$ is the generating function of 3D partitions counting the partitions in an anisotropic way\cite{IKV}:
\begin{eqnarray}\displaystyle
C_{\lambda\,\mu\,\nu}(t,q)\times M(t,q)=\sum_{\pi(\lambda\,\mu\,\nu)}q^{|\pi(\lambda\,\mu\,\nu)|_{\nu}}\,t^{|\pi(\lambda\,\mu\,\nu)|-|\pi(\lambda\,\mu\,\nu)|_{\nu}},
\end{eqnarray}
where $\pi(\lambda\,\mu\,\nu)$ is 3D partition with asymptotics determined by $\lambda,\mu$ and $\nu$ along the three directions and $|\pi(\lambda\,\mu\,\nu)|$ is the regularized total number of boxes in $\pi(\lambda\,\mu\,\nu)$ \cite{ORV}. The shape of the partition $\nu$ along the z-axis determines a slicing of the 3D partition by planes parallel to the z-axis. For the usual vertex this slicing of the 3D partition is just a convenient way of being able to calculate the generating function and the result at the end is independent of this slicing. For the refined vertex case, however, this is not the case and the direction along which we slice breaks the cyclic symmetry of the vertex. This direction along which we slice we will call the preferred direction. Some slices are labelled by $q$ and some slices are labelled by $t$ depending on the shape of the partition $\nu$.  $|\pi(\lambda\,\mu\,\nu)|_{\nu}$ is the number of boxes that fall on the $q$-slices. Each slice is assigned one of the two counting parameters $q$ or $t$ depending on whether this particular slice is located between an inner,$\{v_{i}\}$, and outer corner ,$\{u_{i}\} $, or between an outer and an inner while we trace the profile of $\nu$, \figref{slicing}.
\begin{figure}[h]
  \centering

  \includegraphics[width=2.5in]{slicing.eps}\\
  \caption{The counting scheme is uniquely determined by the representation along the preferred direction.}\label{slicing}
\end{figure}

If all representations labelling the vertex are trivial, the counting is depicted in \figref{macmahon}, and the refined MacMahon function is the generating function of 3D partition with respect to this counting.

\begin{figure}[h]
  \centering

  \includegraphics[width=2.4in]{slicing3d.eps}\\
  \caption{Red slices (left) are counted with $t$ and the blue slices (right) with $q$.}\label{macmahon}
\end{figure}

For a any partition $\nu$ (along the z-axis), it is clear that slices in the $x$ and $y$ direction(far enough) are counted with either $q$ alone or  counted with $t$ alone.  Thus we see that the two un-preferred directions are either associated with $t$ or with $q$, \textit{i.e.}, with $\epsilon_{1}$ or $\epsilon_{2}$.

The A-model topological string theory on a Calabi-Yau threefold $X$ is shown to be the same as the M-theory partition function on \cite{DVV}
\begin{eqnarray}\displaystyle
X\times S^{1}\times TN,
\end{eqnarray}
where $TN$ is the Taub-NUT space. The lift to M-theory provides a novel way to understand the topological A-branes. The A-branes are the M5 branes which wrap the half dimensional Lagrangian submanifolds $L$ in the internal space $X$ and extend in ${\mathbb C}\times S^{1}\subset TN\times S^{1}$. The Taub-NUT space has two cigar subspaces, in the usual topological string theory they can not be distinguished. However, for generic values of  $\epsilon_{1,2}$ we have two different of branes \cite{ACDKV,AS}. The different type of branes are related to the choice of which un-preferred direction we wrap the A-branes. In other words, the open topological string amplitudes for a stack of A-branes on different un-preferred directions are not the same, unlike in the unrefined topological strings.

\begin{figure}[h]
  \centering

  \includegraphics[width=3in]{glue.eps}\\
  \caption{Gluing of two refined vertices}\label{glue}
\end{figure}

When gluing two refined vertices together along an un-preferred direction we must have the edge of one vertex associated with $\epsilon_{1}$ and the edge of the second vertex associated with $\epsilon_{2}$, as shown in \figref{glue} the symmetry between $q$ and $t$ in the refined partition function, which should hold on general grounds Eq.(\ref{sym}), is lost.

Also since at each vertex we have a preferred edge, along which we slice in order to define the refined vertex, in the web diagram we will have a number of preferred edges (one for each vertex). Unless all these preferred edges are parallel to each other, i.e., we are slicing all the 3D partitions that appear at the vertices in the same way, the different vertices can not be glued correctly. Hence, a very important constraint of the refined vertex is that all preferred edges in the web diagram should be parallel. As we will discuss in the next section this condition can not always be satisfied and, therefore, refined vertex formalism only works for a certain class of geometries. A web diagram will have set of parallel preferred edges if the corresponding Calabi-Yau threefold is a $A_n$ fibration over a $\mathbf{P}^1$ or a chain of $\mathbf{P}^1$'s.

\section{The case of local $\mathbf{P}^2$}

In order to calculate the refined partition function of this geometry we need choose at every vertex of the web diagram an edge which is associated with the direction we used to slice the 3D partition. This choice is arbitrary for a given vertex, however, all such edges in the web diagram need to be parallel to each other. This essentially means that we need a set of edges of the web diagram such that this set covers all the vertices and all  edges in this set are parallel. It is easy to see that many web diagram fail to have such a set of edges and therefore refined vertex formalism can not be directly applied to such web diagrams.

The simplest geometry which can not be dealt with using the refined topological vertex is ${\cal O}(-3) \mapsto \mathbf{P}^{2}$ also referred to as local $\mathbf{P}^{2}$. The Newton polygon and the web diagram of this geometry is given in \figref{p2} below.

\begin{figure}
  \centering

  \includegraphics[width=4in]{p2.eps}\\
  \caption{The Newton polygon and the web diagram of local $\mathbf{P}^{2}$}\label{p2}
\end{figure}

Recall from the discussion in the last section that the refined vertex is the open string amplitude for branes (along the un-preferred direction) which fill the $z_{1}$-plane and the $z_{2}$-plane \textit{i.e.}, the two branes on the two un-preferred legs of the vertex are of different kind, one of them is $\epsilon_{1}$-brane and the other is the $\epsilon_{2}$-brane. If we choose one of the internal line of the web diagram of local $\mathbf{P}^{2}$ to be the preferred edge (as shown in \figref{p22}) then the refined vertex factor associated with this edge is given by
\begin{eqnarray}\displaystyle\label{eq1}
C_{\lambda^t\,\emptyset\,\nu}(t,q)\,C_{\emptyset\,\mu\,\nu^t}(q,t)\sim \widetilde{Z}_{\nu}(t,q)\widetilde{Z}_{\nu^t}(q,t)\,\Big[s_{\lambda}(t^{-\rho}q^{-\nu})\,s_{\mu}(t^{-\rho}q^{-\nu})\Big]
\end{eqnarray}
\begin{figure}[h]
  \centering

  \includegraphics[width=1.5in]{p22.eps}\\
  \caption{The refined topological vertices can not be consistently glued with the respect to the rules of refined vertex.}\label{p22}
\end{figure}

It follows from the factor in the square bracket in Eq. (\ref{eq1}) that the brane on the left lower vertex associated with partition $\lambda$ and the brane associated with partition $\mu$ on the right lower vertex are both $\epsilon_{2}$-brane. Therefore this suggests that the slicing of the local $\mathbf{P}^2$ web diagram by $\epsilon_{\alpha}$-branes is such that upper vertex in \figref{p22} has $\epsilon_{1}$-branes on both its internal legs as shown in \figref{localp2}.

\begin{figure}[h]
  \centering

  \includegraphics[width=3in]{localp2.eps}\\
  \caption{Slicing of the local ${\bf P}^2$ web diagram by $\epsilon_{\alpha}$-branes.}\label{localp2}
\end{figure}

Thus we see that the upper vertex has a different brane configuration then the lower ones which had different $\epsilon_{\alpha}$ branes on there un-preferred edges. Thus we need a new refined topological vertex in which the branes on the un-preferred edges should fill the same plane in the spacetime. Thus to complete the refined vertex formalism and to calculate the refined partition function of local $\mathbf{P}^{2}$ we need two refined topological vertices corresponding to two different brane configurations as shown in \figref{twovertex}.

\begin{figure}[h]
  \centering

  \includegraphics[width=3in]{twovertex.eps}\\
  \caption{Brane configurations corresponding to two distinct refined vertices. The label $\epsilon_{1,2}$ on the brane indicates which subspace of $\mathbb{C}^2$ it fills.}\label{twovertex}
\end{figure}

\subsection{The refinement of local $\mathbf{P}^{2}$}

In this section, we will calculate the refined partition function of local ${\bf P}^2$ and, along the way, determine the two-legged version of the new refined vertex.

We will begin with local $\mathbf{P}^{2}$ blown up at one point. The blowup of $\mathbf{P}^{2}$ is a Hirzebruch surface \textit{i.e.}, it is a $\mathbf{P}^{1}$ bundle over $\mathbf{P}^{1}$ such that locally this bundle is ${\cal O}(-n)$ over $\mathbf{P}^{1}$. The local $\mathbf{F}_{1}$ is then the total space of the canonical bundle on $\mathbf{F}_{1}$. The Newton polygon and the web diagram of local $\mathbf{F}_{1}$ is shown in \figref{hirzebruch1}.

\begin{figure}[h]
  \centering

  \includegraphics[width=3in]{hirzebruch1.eps}\\
  \caption{The Newton polygon and the web diagram of local ${\bf F}_{1}$.}\label{hirzebruch1}
\end{figure}

$B$ and $F$ are the base and the fiber $\mathbb{P}^{1}$ respectively and satisfy have the following intersection numbers,
\begin{eqnarray}\displaystyle
B\cdot B=-1\,,\,\,\,F\cdot F=0\,,\,\,\,\,B\cdot F=+1\,.
\end{eqnarray}
For a $\mathbf{P}^{1}$ embedded in a Calabi-Yau threefold the local geometry is ${\cal O}(m)\oplus {\cal O}(-2-m)\mapsto \mathbf{P}^{1}$ for some $m$. In this $B$ is an exceptional curve coming from the blowup of $\mathbf{P}^{2}$ and has local geometry ${\cal O}(-1)\oplus {\cal O}(-1)\mapsto B$ i.e., it is resolved conifold. thus the curve $B$ can undergo flop transition which is also clear from the Newton polygon in \figref{hirzebruch1} which has two distinct triangulations. The flop transition of $B$ can take us from local $\mathbf{F}_{1}$ to local $\mathbf{P}^{2}$ as shown in \figref{flop}.

\begin{figure}[h]
\begin{center}
\includegraphics[width=3in]{flop.eps}
\end{center}
\caption{The flop of local ${\bf F}_{1}$ gives rise to local ${\bf P}^{2}$ when the flopped curve is taken to infinite size}\label{flop}
\end{figure}

The curve $E$ comes from the flop of $B$ and $H$ is the hyperplane class of $\mathbf{P}^{2}$ and is given by $B+F$. The relation between K\"ahler parameters on the two sides of the flop is given by
\begin{eqnarray}\displaystyle
Q=Q_{b}^{-1}\,,\,\,\,Q_{H}=Q_{b}Q_{f}\,,
\end{eqnarray}
where $Q=e^{-t_{E}}$ is the parameter associated with the curve $E$.

From \figref{flop} it is clear that we can obtain the refined partition function of local $\mathbf{P}^{2}$ from the refined partition function of local $\mathbf{F}_{1}$, $Z_{local\, \mathbf{F}_{1}}(Q_{b},Q_{f},t,q)$, if we can follow flop transition in the partition functions:
\begin{eqnarray}\displaystyle
Z_{local \,\mathbf{P}^{2}}(Q_{H},t,q)=\mbox{lim}_{Q_{E}\mapsto 0}\, Z_{local \,\mathbf{F}_{1}}(Q_{E}^{-1},Q_{H}Q_{E},t,q)
\end{eqnarray}

The refined partition function of the local $\mathbf{F}_{1}$ can be calculated using the refined topological vertex, the two internal horizontal line in the web diagram of the local $\mathbf{F}_{1}$ (\figref{hirzebruch1}) can serve as the preferred edges, and is given by \cite{IKV},
\begin{align}\nonumber
Z_{{\bf F}_{1}}(Q_{b},Q_{f}):&=\sum_{\nu\,\lambda\,\mu\,\sigma}(-Q_{b}Q_{f})^{|\nu|}(-Q_b)^{|\sigma|}(-Q_{f})^{|\lambda|+|\mu|}\,f_{\nu}^{2}(q,t)\,\widetilde{f}_{\lambda}(t,q)\,\widetilde{f}_{\mu}(q,t)\\
&\times C_{\emptyset\, \lambda\,\nu^{t}}(t,q)\,C_{\mu^{t}\,\emptyset\,\nu}(q,t)\,C_{\lambda^{t}\,\emptyset\,\sigma}(t,q)\,C_{\emptyset\,\mu\,\sigma^{t}}(q,t),
\end{align}
where we are using the following convention for the framing factors\footnote{This convention eliminates the need for shifting certain K\"ahler parameters by $\frac{\epsilon_{1}+\epsilon_{2}}{2}$.}
\begin{align}\nonumber
f_{\lambda}(t,q)=(-1)^{|\lambda|}\,t^{\frac{\Arrowvert\lambda^t\Arrowvert^2}{2}}\,q^{-\frac{\Arrowvert\lambda\Arrowvert^2}{2}}\,,\,\,\,\,\,\widetilde{f}_{\lambda}(t,q)=(-1)^{|\lambda|}\,t^{\frac{\Arrowvert\lambda^t\Arrowvert^2-|\lambda|}{2}}\,q^{-\frac{\Arrowvert\lambda\Arrowvert^2-|\lambda|}{2}}.
\end{align}
After the flop transition the relevant variables are $Q=Q_{b}^{-1}$ and $Q_{H}=Q_{b}Q_{f}$. In terms of these variables the partition function is
\begin{align}\nonumber
Z^{flopped}_{{\bf F}_{1}}(Q_{H},Q)&=\sum_{\nu\,\lambda\,\mu\,\sigma}(-Q_{H})^{|\lambda|+|\mu|+|\nu|}(-Q)^{-|\sigma|}\,Q^{|\lambda|+|\mu|}\,f_{\nu}^{2}(q,t)\,\widetilde{f}_{\lambda}(t,q)\,\widetilde{f}_{\mu}(q,t)\\
&\times C_{\emptyset\, \lambda\,\nu^{t}}(t,q)\,C_{\mu^{t}\,\emptyset\,\nu}(q,t)\,C_{\lambda^{t}\,\emptyset\,\sigma}(t,q)\,C_{\emptyset\,\mu\,\sigma^{t}}(q,t).
\end{align}

Separating the sum involving $Q$, we can write the partition function as
\begin{align}\nonumber
Z^{flopped}_{{\bf F}_{1}}(Q_{H},Q)&=\sum_{\nu\,\lambda\,\mu\,\sigma}(-Q_{H})^{|\lambda|+|\mu|+|\nu|}\,f_{\nu}^{2}(q,t)\,\widetilde{f}_{\lambda}(t,q)\,\widetilde{f}_{\mu}(q,t)\,C_{\emptyset\, \lambda\,\nu^{t}}(t,q)\,C_{\mu^{t}\,\emptyset\,\nu}(q,t)\\\label{f1}
&\times\,\, Q^{|\lambda|+|\mu|}\,Z_{\lambda\,\mu}(Q^{-1},t,q),
\end{align}
where we have defined
\begin{align}\label{rhl}
Z_{\lambda\,\mu}(Q,t,q)\coloneqq\sum_{\sigma}(-Q)^{|\sigma|}\,C_{\lambda^{t}\,\emptyset\,\sigma}(t,q)\,C_{\emptyset\,\mu\,\sigma^{t}}(q,t).
\end{align}
In the limit $Q\mapsto 0$, Eq. (\ref{f1}) gives the partition function of the local ${\bf P}^{2}$,
\begin{align}\label{p2pf}
Z_{{\bf P}^{2}}(Q_{H})=\lim_{Q\mapsto 0}\,Z^{flopped}_{{\bf F}_{1}}(Q_{H},Q)\,.
\end{align}
The $Q$ dependent factor in Eq. (\ref{f1}) can be calculated exactly. We leave the detailed calculation for the next section and here just give the result:
\begin{align}\label{f2}
\lim_{Q\mapsto 0}\,Q^{|\lambda|+|\mu|}\,Z_{\lambda\,\mu}(Q^{-1})=\left(\frac{q}{t}\right)^
{\frac{|\lambda|+|\mu|}{2}}(-1)^{|\mu|}f_{\mu^t}(t,q)
\sum_{\eta\,\,\sigma}N^{\eta}_{\lambda\,\mu}U_{\eta\,\sigma}(q,t)\,
f_{\sigma}(t,q)\,P_{\sigma}(t^{-\rho};q,t),
\end{align}
where in the above equation $N^{\eta}_{\lambda\,\mu}$ are the Littlewood-Richardson coefficients, $P_{\sigma}({\bf x};q,t)$ are the Macdonald polynomials and $U_{\eta\,\sigma}(q,t)$ are the transformation matrices which transform Macdonald polynomials to Schur polynomials, $s_{\eta}({\bf x})=\sum_{\sigma}U_{\eta\,\sigma}(q,t)P_{\sigma}({\bf x};q,t)$.

Thus the refined partition function of locall $\mathbf{P}^{2}$ is given by
\begin{eqnarray}\displaystyle\label{eq37}
Z_{local \,\mathbf{P}^2}(Q,t,q)&=&\sum_{\lambda\,\mu\,\nu}(-Q)^{|\lambda|+|\mu|+|\nu|}\,q^{\frac{3||\nu^t||^2}{2}}\,t^{-\frac{||\nu||^2}{2}}\,\widetilde{Z}_{\nu}(q,t)
 \widetilde{Z}_{\nu^t}(t,q)\times\\\nonumber&&s_{\lambda}(q^{-\rho}t^{-\nu})\,s_{\mu}(q^{-\rho}t^{-\nu})\,\Big(\frac{q}{t}\Big)^{\frac{|\lambda|-|\mu|}{2}}\,N^{\eta}_{\lambda\,\mu}
 \Big[\sum_{\sigma}\,U_{\eta\,\sigma}t^{\frac{||\eta^t||^2}{2}}q^{-\frac{||\eta||^2}{2}}\,P_{\sigma}(t^{-\rho};q,t)\Big]\,.
\end{eqnarray}
Define\begin{eqnarray}\displaystyle\nonumber
T_{\mu\, \lambda\,\emptyset}(t,q)&\coloneqq&(-1)^{|\mu|}\left(\frac{t}{q}\right)^{|\lambda|}\,f_{\lambda}(q,t)\,f^{2}_{\mu^t}(t,q)\,
\sum_{\eta\,\,\sigma}N^{\eta}_{\lambda^t\,\mu}U_{\eta\,\sigma}(q,t)\,
f_{\sigma}\,P_{\sigma}(t^{-\rho};q,t)
\end{eqnarray}
Then from Eq. (\ref{eq37}) it follows that
\begin{eqnarray}\displaystyle
\label{eee}
Z_{{\mathbf P}^{2}}(Q_{H})&=&\sum_{\lambda\,\mu\,\nu}(-Q_{H})^{|\lambda|+|\mu|+|\nu|}\,
\Big(f_{\nu}(q,t)\,\widetilde{f}_{\lambda}(t,q)\,\widetilde{f}_{\mu}(q,t)\Big)^{2}\\\nonumber
&&\,C_{\emptyset\,\lambda\,\nu^t}(t,q)C_{\mu^t\,\emptyset\,\nu}(q,t)\,T_{\mu\,\lambda^t\,\emptyset}
\end{eqnarray}
We have verified, up to order $Q_{H}^3$, that the above partition function gives the correct BPS degeneracies for local ${\bf P}^{2}$. The form of the partition function given above suggests that $T_{\lambda\,\mu\,\emptyset}(t,q)$ is a new two-legged topological vertex which can be used to determine the refined partition function of the local ${\bf P}^{2}$ using the usual gluing rule.

\section{The New Refined Vertex}

In this section, we derive the new refined vertex by placing the branes on the resolved conifold and then letting the geometry undergo a flop transition. The brane configuration on the resolved conifold that we will begin with corresponds to the Hopf link and was studied in \cite{GIKV}. The relation of this brane configuration and its corresponding amplitude with the refined Chern-Simons theory was discussed in \cite{IK}. In the next section we will clarify further the relation between the refined topological vertex, the new vertex and the refined Chern-Simons theory. We will see that the refined topological vertex and the new vertex together capture all the information of the refined Chern-Simons theory.

Let us begin, as is always the case when considering knots in the open topological string theory, with $T^{*}S^{3}$, the cotangent of $S^3$. Consider the Hopf link embedded in $S^{3}$ which is the zero section of $T^{*}S^{3}$. The two unknots which make up the Hopf link are colored by two representations $\lambda$ and $\mu$.  When this setup is embedded in M-theory we consider compactification on $T^{*}S^{3}\times S^{1}\times \mathbb{C}^2$ with M5-branes wrapping  $L_{1}\times S^{1}\times C_{1}$ and $L_{2}\times S^{1}\times C_{2}$. $L_{1,2}$ are two Lagrangian cycles with boundary the two unknots of the Hopf link. $C_{1,2}$ are 2 dimensional planes in $\mathbb{C}^2$ given by $z_{1}=0$ and $z_{2}=0$ respectively. In the refined topological string theory, two choices for the  branes on Lagrangian cycles: either both M5-branes fill $z_{1}$-plane (or $z_{2}$-plane) or one brane fills the $z_{1}$-plane and the other $z_{2}$-plane as shown in \figref{a1}. We will denote these two configurations with ${\cal C}_{1}$ and ${\cal C}_{2}$, respectively.

\begin{figure}[h]
  \centering

  \includegraphics[width=3.3in]{a1.eps}\\
  \caption{Two distinct brane configurations on $T^{*}S^3$.}\label{a1}
\end{figure}

When the $T^{*}S^3$ undergoes geometric transition $S^{3}$ shrinks to zero size and is replaced with an $S^{2}$. The total space becomes a complex bundle over $\mathbf{P}^1$, ${\cal O}(-1)\oplus {\cal O}(-1)\mapsto \mathbf{P}^1$. The two configurations ${\cal C}_{1,2}$, after the geometric transition, correspond to the two A-brane configurations shown in \figref{config}.

\begin{figure}[h]
  \centering

  \includegraphics[width=3in]{config.eps}\\
  \caption{In the refined open topological string theory, the branes are distinguished with respect to which $\mathbb{R}^{2}\subset \mathbb{R}^{4}$ they fill.}\label{config}
\end{figure}

The two different configurations of branes given above give two different partition functions. The brane configuration in ${\cal C}_{1}$ is the same brane configuration for the Hopf link studied in \cite{GIKV}. In this brane configuration, the M5-branes wrapping both Lagrangian submanifolds on the different legs of the resolved conifold fill the $z_{1}$-plane in ${\mathbb C}^{2}$. The BPS particles due to M2-branes wrapping the discs ending on M5-branes carry spin under the $U(1)$ rotation of the $z_{1}$-plane (they also carry $U(1)$ R-charge which is the $U(1)$ rotation of the orthogonal $z_{2}$-plane). The BPS particles correspond to holomorphic functions on the $z_{1}$-plane and therefore the open string partition function of this brane configuration will have denominator a function of $q$ only. However, in ${\cal C}_{2}$, the branes are extending in both $z_{1}$- and $z_{2}$-planes, and depending on which discs M2-branes are wrapping, the BPS particles are charged under both $U(1)$'s and the denominator of the open string partition function in this case will be function of both $t$ and $q$.

\subsection{Derivation of the two-legged new vertex}
In this subsection, we will present the derivation of the new vertex. This subsection and the next one include the more technical details. We collect some of the identities used in the derivation in Appendix B.

The configuration ${\cal C}_{1}$  after the flop corresponds to the refined Hopf link \cite{GIKV} and the corresponding open string partition function is given by,
\begin{eqnarray}\displaystyle\label{hopflink}
Z_{\lambda\,\mu}(Q,t,q)=\sum_{\nu}(-Q)^{|\nu|}\,C_{\lambda^{t}\,\emptyset\,\nu}(t,q)\,C_{\emptyset\,\mu\,\nu^{t}}(q,t),
\end{eqnarray}
where $T=-\mbox{log}(Q)$ is the K\"ahler parameter of the ${\bf P}^{1}$.

\begin{figure}[h]
  \centering

\includegraphics[width=3in]{hopf.eps}\\
  \caption{The relation between the Hopf link and the legged version of the new vertex.}\label{hopf}
\end{figure}

As shown in \figref{hopf} the configuration ${\cal C}_{1}$ of branes is precisely what appears in the local ${\bf F}_{1}$. Thus if we can follow the flop in the refined Hopf link partition function we can determine the two-legged version of the new vertex which appeared in Eq.(\ref{eee}).

After the flop the corresponding partition function will be (up to framing factors), in terms of the new vertex $T_{\lambda\,\mu\,\nu}(t,q)$,
\begin{eqnarray}\displaystyle
Z_{\lambda\,\mu}(Q^{-1},t,q)\sim \sum_{\nu}Q^{-|\nu|}\,T_{\lambda\,\mu\,\nu}(t,q)T_{\emptyset\,\emptyset\,\nu^t}(q,t)\,.
\end{eqnarray}
We will see that the expression for $T_{\lambda\,\mu\,\nu}(t,q)$ can be read from the partition function Eq. (\ref{hopflink}) by analytic continuation $Q$ to $Q^{-1}$ so that
\begin{eqnarray}\displaystyle
Z_{\lambda\,\mu}^{flop}(Q,t,q)\sim Z_{\lambda\,\mu}(Q^{-1},t,q)\,.
\end{eqnarray}

Although $Z_{\lambda\,\mu}(Q,t,q)$ is an infinite series in $Q$, the normalized open string partition function
\begin{eqnarray}\displaystyle
G_{\lambda\,\mu}(Q)\coloneqq\frac{Z_{\lambda\,\mu}(Q,t,q)}{Z_{\emptyset\,\emptyset}(Q,t,q)}=\frac{\sum_{\sigma}(-Q)^{|\sigma|}\,C_{\lambda^{t}\,\emptyset\,\sigma}(t,q)\,C_{\emptyset\,\mu\,\sigma^{t}}(q,t)}{\sum_{\sigma}(-Q)^{|\sigma|}\,C_{\emptyset\,\emptyset\,\sigma}(t,q)\,C_{\emptyset\,\emptyset\,\sigma^{t}}(q,t)}
\end{eqnarray}
is a polynomial of degree $|\lambda|+|\mu|$ in $Q$. This was conjectured in \cite{GIKV} and proven in \cite{AK}. Notice that $Z_{\emptyset\,\emptyset}(Q)$ is the closed string partition function of the geometry ${\cal O}_{\mathbb{P}^{1}}(-1)\oplus{\cal O}_{\mathbb{P}^{1}}(-1)$ and is invariant under the flop,\footnote{It is invariant if we ignore the shift in the classical contribution which arises due to the flop.}
\begin{eqnarray}\displaystyle
Z_{\emptyset\,\emptyset}(Q^{-1},t,q)=Z_{\emptyset\,\emptyset}(Q,t,q)\,.
\end{eqnarray}
Thus we can obtain the expression for the flopped partition function by studying $G_{\lambda\,\mu}(Q,t,q)$. Using the definition of the refined topological vertex we can write $G_{\lambda\,\mu}(Q,t,q)$ as
\begin{eqnarray}\displaystyle\nonumber
G_{\lambda\,\mu}(Q)&=&(-1)^{|\mu|}\left(\frac{q}{t}\right)^{\frac{|\lambda|+|\mu|}{2}}f_{\mu^t}\frac{\sum_{\nu}\,(-Q)^{|\nu|}\,q^{\frac{\Arrowvert\nu\Arrowvert^2}{2}}\,t^{\frac{\Arrowvert\nu^t\Arrowvert^2}{2}}\,\widetilde{Z}_{\nu}(t,q)\,\widetilde{Z}_{\nu^t}(q,t)\,s_{\lambda}(t^{-\rho}q^{-\nu})\,s_{\mu}(t^{-\rho}q^{-\nu})}{\sum_{\nu}\,(-Q)^{|\nu|}\,q^{\frac{\Arrowvert\nu\Arrowvert^2}{2}}\,t^{\frac{\Arrowvert\nu^t\Arrowvert^2}{2}}\,\widetilde{Z}_{\nu}(t,q)\,\widetilde{Z}_{\nu^t}(q,t)}\,.
\end{eqnarray}
Since the two Schur functions in the summation above have the same argument we can use
\begin{eqnarray}\displaystyle
s_{\lambda}({\bf x})\,s_{\mu}({\bf x})=\sum_{\eta}N^{\eta}_{\lambda\,\mu}s_{\eta}({\bf x}),
\end{eqnarray}
to write the normalized open amplitude as
\begin{align}
G_{\lambda\,\mu}(Q,t,q)&=(-1)^{|\mu|}\left(\frac{q}{t}\right)^{\frac{|\lambda|+|\mu|}{2}}f_{\mu^t}(t,q) \frac{1}{Z_{\emptyset\,\emptyset}}\sum_{\eta}\,N^{\eta}_{\lambda\,\mu}\,H_{\eta}(Q,t,q)\\\nonumber
H_{\eta}(Q,t,q)&\coloneqq\sum_{\nu}\,(-Q)^{|\nu|}\,q^{\frac{\Arrowvert\nu\Arrowvert^2}{2}}\,t^{\frac{\Arrowvert\nu^t\Arrowvert^2}{2}}\,\widetilde{Z}_{\nu}(t,q)\,\widetilde{Z}_{\nu^t}(q,t)\,
s_{\eta}(t^{-\rho}q^{-\nu}).
\end{align}
Following \cite{AK}, we express the Schur function in terms of Macdonald function to obtain
\begin{eqnarray}\displaystyle
H_{\eta}(Q,t,q)&=&\sum_{\sigma}U_{\eta\,\sigma}(q,t)J_{\sigma}(Q,t,q)\\\nonumber
J_{\sigma}(Q,t,q)&\coloneqq&\sum_{\nu}\,(-Q)^{|\nu|}\,q^{\frac{\Arrowvert\nu\Arrowvert^2}{2}}\,t^{\frac{\Arrowvert\nu^t\Arrowvert^2}{2}}\,\widetilde{Z}_{\nu}(t,q)\,\widetilde{Z}_{\nu^t}(q,t)\,
P_{\sigma}(t^{-\rho}q^{-\nu};q,t),
\end{eqnarray}
where the $U_{\eta\,\sigma}(q,t)$ are the previously defined matrices which take Macdonald functions to Schur functions.
$s_{\eta}({\bf x})$ and $P_{\sigma}({\bf x};q,t)$ are both homogenous functions of degree of $|\eta|$ and $|\sigma|$, respectively, hence, $U_{\eta\,\sigma}(q,t)=0$ if $|\eta|\neq |\sigma|$. Also it follows from the definition of the Macdonald function that $U_{\eta\,\sigma}=\delta_{\eta\,\sigma}$ for $t=q$.\\

The normalized Hopf link partition function in terms of $J_{\sigma}$ is given by
\begin{eqnarray}\displaystyle
G_{\lambda\,\mu}(Q)=(-1)^{|\mu|}\left(\frac{q}{t}\right)^{\frac{|\lambda|+|\mu|}{2}}f_{\mu^t}(t,q)\frac{1}{Z_{\emptyset\,\emptyset}(Q)}\sum_{\eta\,\,\sigma}N^{\eta}_{\lambda\,\mu}\,U_{\eta\,\sigma}(q,t)\,J_{\sigma}(Q,t,q).
\end{eqnarray}
The principal specialization of the Macdonald polynomials are proportional to $\widetilde{Z}_{\nu}(t,q)$
\begin{eqnarray}\displaystyle
t^{\frac{\Arrowvert\nu^t\Arrowvert^2}{2}}\,\widetilde{Z}_{\nu}(t,q)&=&P_{\nu}(t^{-\rho};q,t),
\end{eqnarray}
$J_{\sigma}(Q,t,q)$ can be written in terms of Macdonald functions
\begin{eqnarray}\displaystyle
J_{\sigma}(Q,t,q)&=&\sum_{\nu}\,(-Q)^{|\nu|}\,\,P_{\nu^t}(q^{-\rho};t,q)\,P_{\nu}(t^{-\rho};q,t)\,
P_{\sigma}(t^{-\rho}q^{-\nu};q,t).\\\nonumber
\end{eqnarray}
The sum of the partition $\nu$ can be performed by making use of the identity (\cite{macdonald}, Page 332, Eq.(6.6))
\begin{eqnarray}\displaystyle
P_{\nu}(t^{-\rho};q,t)P_{\sigma}(q^{-\nu}\,t^{-\rho};q,t)=P_{\sigma}(t^{-\rho};q,t)\,P_{\nu}(q^{-\sigma}\,t^{-\rho};q,t),
\end{eqnarray}
$J_{\sigma}(Q,t,q)$ takes a product form
\begin{eqnarray}\displaystyle\nonumber
J_{\sigma}(Q,t,q)&=&\sum_{\nu}\,(-Q)^{|\nu|}\,P_{\nu^t}(q^{-\rho};t,q)\,P_{\sigma}(t^{-\rho};q,t)\,P_{\nu}(t^{-\rho}q^{-\sigma};q,t)\\\nonumber
&=&P_{\sigma}(t^{-\rho};q,t)\sum_{\nu}\,(-Q)^{|\nu|}\,P_{\nu^t}(q^{-\rho};t,q)\,P_{\nu}(t^{-\rho}q^{-\sigma};q,t)\\
&=&P_{\sigma}(t^{-\rho};q,t)\,\prod_{i,j=1}^{\infty}\left(1-Q\,t^{-\rho_{i}}\,q^{-\rho_{j}-\sigma_{i}}\right),\\\nonumber
\end{eqnarray}
where we have used the identity \cite{macdonald}
\begin{eqnarray}\displaystyle
\sum_{\nu}\,P_{\nu}({\bf x};q,t)\,P_{\nu^t}({\bf y};t,q)=\prod_{i,j=1}^{\infty}\left(1+x_{i}\,y_{j}\right).
\end{eqnarray}
We can factorize the infinite product as a finite product over the boxes of a Young diagram and an infinite one to normalize $J_{\sigma}(Q,t,q)$:
\begin{align}
&J_{\sigma}(Q,t,q)=\left[P_{\sigma}(t^{-\rho};q,t)\,\prod_{(i,j)\in \sigma}(1-Q\,t^{i-\frac{1}{2}}\,q^{-j+\frac{1}{2}})\right]\times \prod_{i,j=1}^{\infty}\left(1-Q\,t^{i-\frac{1}{2}}\,q^{j-\frac{1}{2}}\right)\\\nonumber
&\frac{J_{\sigma}(Q,t,q)}{J_{\emptyset}(Q,t,q)}=P_{\sigma}(t^{-\rho};q,t)\,\prod_{(i,j)\in \sigma}(1-Q\,t^{i-\frac{1}{2}}\,q^{-j+\frac{1}{2}}).
\end{align}
$J_{\emptyset}(Q,t,q)=Z_{\emptyset\,\emptyset}(Q,t,q)$ is the closed string partition function for the resolved conifold. Thus the normalized refined Hopf link partition function can be written as\footnote{The normalized Hopf link partition function can be even written in a more compact way realizing that
\begin{eqnarray}\displaystyle
P_{\sigma}(t^{-\rho},z\,t^{\rho};q,t)=P_{\sigma}(t^{-\rho};q,t)\prod_{(i,j)\in \sigma}(1-z\,q^{j-1}\,t^{-i+1}).
\end{eqnarray}
$G_{\lambda\,\mu}(Q,t,q)$ takes then the following form
\begin{eqnarray}\displaystyle\nonumber
G_{\lambda\,\mu}(Q)&=&(-1)^{|\mu|}\left(Q\sqrt{\frac{q}{t}}\right)^{|\lambda|+|\mu|}
f_{\mu^t}(t,q)\sum_{\eta\,\,\sigma}N^{\eta}_{\lambda\,\mu}U_{\eta\,\sigma}(q,t)\,f_{\sigma}(t,q)\,P_{\sigma}\left(t^{-\rho},Q^{-1}\sqrt{\frac{q}{t}}\,t^{\rho};q,t\right).
\end{eqnarray}}

\begin{eqnarray}\displaystyle\label{xxxx}\nonumber
G_{\lambda\,\mu}(Q)=(-1)^{|\mu|}\left(\frac{q}{t}\right)^{\frac{|\lambda|+|\mu|}{2}}f_{\mu^t}(t,q)\sum_{\eta\,\,\sigma}N^{\eta}_{\lambda\,\mu}U_{\eta\,\sigma}(q,t)\,P_{\sigma}(t^{-\rho};q,t)\,\prod_{(i,j)\in \sigma}(1-Q\,t^{i-\frac{1}{2}}\,q^{-j+\frac{1}{2}}).\\
\end{eqnarray}
Before we continue with the derivation of the new vertex, let us argue that the normalized partition function for the Hopf link is a polynomial in $Q$ of order $|\lambda|+|\mu|$. We know that $N^{\eta}_{\lambda\,\mu}$ is zero unless $|\lambda|+|\mu|=|\eta|$ and $U_{\eta\,\sigma}(t,q)$ vanishes for $|\eta|\neq|\sigma|$. Therefore, we conclude that the sum is finite, and the product is of finite order. $G_{\lambda\,\mu}(Q,t,q)$ is of the order $|\lambda|+|\mu|$.

The two legged new vertex will be obtained by taking the flop transition and sending the K\"{a}hler parameter to infinity. Let us define $Q_{\bullet}=Q^{-1}$ to take the limit after the flop transition, we can write
\begin{eqnarray}\displaystyle
\prod_{(i,j)\in \sigma}\Big(1-Q\,t^{i-\frac{1}{2}}\,q^{-j+\frac{1}{2}}\Big)&=&\prod_{(i,j)\in \sigma}\Big(1-Q_{\bullet}^{-1}\,t^{i-\frac{1}{2}}\,q^{-j+\frac{1}{2}}\Big)\\\nonumber
&=&(-Q_{\bullet})^{-|\sigma|}\,t^{\frac{\Arrowvert\sigma^t\Arrowvert^2}{2}}\,q^{-\frac{\Arrowvert\sigma\Arrowvert^2}{2}}\,\prod_{(i,j)\in \sigma}\left(1-Q_{\bullet}\,t^{-i+\frac{1}{2}}\,q^{j-\frac{1}{2}}\right)
\end{eqnarray}
The limit to define the two-legged vertex (up to framing and overall factors) can be easily taken using the following form of the normalized Hopf link partition function
\begin{align}\label{xx}\nonumber
G_{\lambda\,\mu}(Q)
&=(-1)^{|\mu|}\left(-Q\sqrt{\frac{t}{q}}\right)^{|\lambda|+|\mu|}f_{\mu^t}(t,q)\sum_{\eta\,\,\sigma}N^{\eta}_{\lambda\,\mu}U_{\eta\,\sigma}(q,t)\,t^{\frac{||\sigma^t||^2}{2}}\,q^{-\frac{||\sigma||^2}{2}}\,P_{\sigma}(t^{-\rho};q,t)\\ &\times\prod_{(i,j)\in \sigma}\left(1-Q^{-1}\,t^{-i+\frac{1}{2}}\,q^{j-\frac{1}{2}}\right).
\end{align}
From the above equation we see that
\begin{eqnarray}\displaystyle
T_{\mu\,\lambda^{t}\,\emptyset}(t,q)&\propto&\lim_{Q\mapsto 0}\,Q^{|\lambda|+|\mu|}\,Z_{\lambda\,\mu}(Q^{-1})=\lim_{Q\mapsto 0}\,Q^{|\lambda|+|\mu|}\,Z_{\emptyset\,\emptyset}(Q^{-1})\,G_{\lambda\,\mu}(Q^{-1})\\\nonumber
&=&(-1)^{|\mu|}\left(\frac{t}{q}\right)^{\frac{|\lambda|}{2}}f_{\mu^t}(t,q)\sum_{\eta\,\,\sigma}N^{\eta}_{\lambda\,\mu}U_{\eta\,\sigma}(q,t)\,f_{\sigma}(t,q)\,P_{\sigma}(t^{-\rho};q,t).
\end{eqnarray}
Notice that we have made use of the invariance of the resolved conifold partition function under flop, $Z_{\emptyset\,\emptyset}(Q^{-1},t,q)=Z_{\emptyset\,\emptyset}(Q,t,q)$

\subsection{The new vertex}
In this subsection, we will derive a three-legged new topological vertex from the flop of the refined Hopf link invariant. Together with the refined topological vertex this new vertex will complete the refined vertex formalism which can then be used for any non-compact toric Calabi-Yau threefold.

From the results of the previous section (Eq. \ref{xx}), we have
\begin{align}\nonumber
G_{\lambda\,\mu}(Q)&=(-1)^{|\mu|}\left(Q\sqrt{\frac{q}{t}}\right)^{|\lambda|+|\mu|}
f_{\mu^t}(t,q)\sum_{\eta\,\,\sigma}N^{\eta}_{\lambda\,\mu}U_{\eta\,\sigma}(q,t)\,f_{\sigma}(t,q)\,P_{\sigma}(t^{-\rho};q,t)\\\nonumber &\times\prod_{(i,j)\in \sigma}\left(1-Q^{-1}\,t^{-i+\frac{1}{2}}\,q^{j-\frac{1}{2}}\right)
\end{align}

Now that we have a polynomial expression for $G_{\lambda\,\mu}$ we can determine the flopped vertex.
\begin{eqnarray}\displaystyle\nonumber
Z_{\emptyset\,\emptyset}(Q^{-1})\prod_{(i,j)\in \sigma}\left(1-Q^{-1}\,t^{-i+\frac{1}{2}}\,q^{j-\frac{1}{2}}\right)&=&\prod_{i,j=1}^{\infty}\left(1-Q^{-1}\,t^{i-\frac{1}{2}}\,q^{j-\frac{1}{2}}\right)\,\,\prod_{(i,j)\in \sigma}\left(1-Q^{-1}\,t^{-i+\frac{1}{2}}\,q^{j-\frac{1}{2}}\right)\\\nonumber
&=&\sum_{\nu}(-Q)^{-|\nu|}P_{\nu^t}(q^{\rho};t,q)\,P_{\nu}(t^{\rho}\,q^{\sigma};q,t)
\end{eqnarray}
Using the identity
\begin{eqnarray}\displaystyle
P_{\nu}(t^{\rho}q^{\sigma};q,t)=\frac{P_{\sigma}(t^{\rho}q^{\nu};q,t)\,P_{\nu}(t^{\rho};q,t)}{P_{\sigma}(t^{\rho};q,t)}
\end{eqnarray}
we get
\begin{eqnarray}\displaystyle\nonumber
Z_{\lambda\,\mu}(Q)&=&(-1)^{|\mu|}\left(Q\sqrt{\frac{q}{t}}\right)^{|\lambda|+|\mu|}f_{\mu^t}(t,q)\sum_{\nu\,\eta\,\sigma}(-Q)^{-|\nu|}\,
N^{\eta}_{\lambda\,\mu}\,U_{\eta\,\sigma}(q,t)\,f_{\sigma}(t,q)\,P_{\nu^t}(q^{\rho};t,q)\,\\\nonumber
&\times&\left(\frac{P_{\sigma}(t^{-\rho};q,t)}{P_{\sigma}(t^{\rho};q,t)}\right)\,P_{\nu}(t^{\rho};q,t)\,P_{\sigma}(t^{\rho}q^{\nu};q,t).
\end{eqnarray}
The ratio in the parentheses can be simplified to a factor proportional to the framing factor
\begin{eqnarray}\displaystyle
\frac{P_{\sigma}(t^{-\rho};q,t)}{P_{\sigma}(t^{\rho};q,t)}=
(-1)^{|\sigma|}\,t^{\frac{\Arrowvert\sigma^t\Arrowvert^2}{2}}\,q^{-\frac{\Arrowvert\sigma\Arrowvert^2}{2}}\,\Big(\frac{q}{t}\Big)^{\frac{|\sigma|}{2}}=
\Big(\frac{q}{t}\Big)^{\frac{|\sigma|}{2}}\,f_{\sigma}(t,q).
\end{eqnarray}
Notice that
\begin{align}\nonumber
Z_{\lambda\,\mu}(Q&)=(-1)^{|\mu|}\left( Q\frac{q}{t}\right)^{|\lambda|+|\mu|}f_{\mu^t}(t,q)\sum_{\nu\,\eta\,\sigma}(-Q)^{-|\nu|}\,N^{\eta}_{\lambda\,\mu}\,U_{\eta\,\sigma}(t,q)\,f_{\sigma}^{2}(t,q)\,P_{\nu^t}(q^{\rho};t,q)\\ \nonumber &\times P_{\nu}(t^{\rho};q,t)\,P_{\sigma}(t^{\rho}q^{\nu};q,t)
\end{align}
Isolating the $\nu$ sum, the preferred direction, we define $\widetilde{C}_{\lambda\,\mu\,\nu}(t,q)$

\begin{eqnarray}\displaystyle
Q^{|\lambda|+|\mu|}\,G_{\lambda\,\mu}(Q^{-1})&=&\sum_{\nu}(-Q)^{|\nu|}P_{\nu^t}(q^{\rho};t,q)\,\widetilde{C}_{\lambda\,\mu\,\nu}(t,q)\\\nonumber
\widetilde{C}_{\lambda\,\mu\,\nu}(t,q)&=&\,(-1)^{|\mu|}\,f_{\mu^t}(t,q)\,P_{\nu}(t^{\rho};q,t)\,\sum_{\eta\,\sigma}N^{\eta}_{\lambda\,\mu}\,U_{\eta\,\sigma}(t,q)\,f_{\sigma}^{2}(t,q)\,P_{\sigma}(t^{\rho}q^{\nu};q,t)
\end{eqnarray}
Define the new topological vertex as
\begin{eqnarray}\displaystyle\nonumber
T_{\lambda\,\mu\,\nu}(t,q)&\coloneqq&\Big(\sqrt{\frac{t}{q}}\Big)^{|\mu|-|\lambda|}\frac{1}{f_{\lambda^t}(t,q)\,f_{\mu}(q,t)}\,\widetilde{C}_{\lambda^t\,\mu\,\nu}(t,q)\end{eqnarray}
After some simplification the new topological vertex can be written as

\begin{eqnarray}\displaystyle
T_{\lambda\,\mu\,\nu}&=&\Big(\sqrt{\frac{t}{q}}\Big)^{|\mu|-|\lambda|}\,(-1)^{|\mu|}\,f_{\mu^t}^{2}(t,q)\,f_{\lambda}(q,t)\,P_{\nu}(t^{\rho};q,t)\,\\\nonumber
&&\sum_{\eta\,\sigma}N^{\eta}_{\lambda^t\,\mu}\,U_{\eta\,\sigma}\,f_{\sigma}^{2}\,P_{\sigma}(t^{\rho}q^{\nu};q,t)
\end{eqnarray}

\section{Gluing Rules for the New Vertex with the Refined Topological Vertex}

In this section, we will describe the algorithm that will allow us to compute the refined topological string partition function for any given toric Calabi-Yau threefold. As argued in \cite{Eynard:2010dh}, any toric Calabi-Yau threefold can be obtained from a large enough fiducial geometry by a series of flop transitions and sending some of the unwanted K\"{a}hler moduli to infinity.
The fiducial geometry that we will use is a ladder diagram, such as the one depicted in the right upper corner in \figref{td}. Each vertex in this diagram locally corresponds to $T^{*}S^3$ where the $S^{3}$ has shrunk. \\ \\
\noindent Consider $T^{*}S^3$ with $N$ M5-branes wrapped on $S^{3}$ and extending in the $z_{2}$-plane along with two stacks of M5-branes on the wrapped on Lagrangian cycles ${\cal L}_{1}$ and ${\cal L}_{2}$, which have the topology of $S^{1}\times \mathbb{R}^{2}$, and extending in the $z_{2}$-plane. The Lagrangian cycles are shown as dotted line in \figref{flop2}. In going through the large $N$ transition the singular geometry, when the $S^{3}$ has shrunk, can resolve itself in two different ways related to each other by a flop as shown in \figref{flop2}.

\begin{figure}[h]
  \centering

  \includegraphics[width=3in]{flop2.eps}\\
  \caption{The large $N$ transition can resolve the conifold in two different ways.}\label{flop2}
\end{figure}
The rules are pretty simple:
\begin{itemize}
\item{Start with the ladder diagram and resolve each node}
\item{Associate the preferred direction of the refined topological vertex every time a node is resolved by $N\mapsto -\infty$}
\item{Associate the preferred direction of the new topological vertex every time a node is resolved by $N\mapsto \infty$}
\item{The assignment of the preferred direction remains unchanged if the flopped edge is not along one of preferred directions}
\item{Flopping an edge along the preferred direction changes the time of the vertex, \textit{i.e.}, if the preferred direction of the refined topological vertex is flopped it become the preferred direction of the new topological vertex, vice versa.}
\item{Assignment of the propagators and $t,q-$identifications are the same as the refined topological vertex}
\end{itemize}
After these rules are applied, it is easy to see that the different type of preferred directions remain parallel among each other. An example with different choices of flops is depicted in \figref{td}.

\begin{figure}[H]
  \centering

  \includegraphics[width=4.5in]{toricdiagrams.eps}\\
  \caption{Different choice of resolutions of the nodes in the ladder diagram and the associated assignments of the preferred directions.}\label{td}
\end{figure}

\section{Applications of the new vertex}
\par{In this section, we want to work out some examples using the new topological vertex.  }
\subsection{Example 1: ${\cal O}_{{\bf P}^{1}}(-1)\oplus {\cal O}_{{\bf P}^{1}}(-1)$}
\par{The first geometry we want to consider is the resolved conifold. Similar to the refined topological vertex computation we can perform the computation in different ways depending on our choice of gluing the vertices. Our first gluing is in the following way }

\begin{eqnarray}\displaystyle
Z&=&\sum_{\nu}(-Q)^{|\nu|}\,T_{\emptyset\,\emptyset\,\nu}(t,q)\,T_{\emptyset\,\emptyset\,\nu^t}(q,t)\\\nonumber
&=&\sum_{\nu}(-Q)^{|\nu|}\,P_{\nu}(t^{\rho};q,t)\,P_{\nu^t}(q^{\rho};t,q)=\prod_{i,j=1}^{\infty}\Big(1-Q\,q^{i-\frac{1}{2}}\,t^{j-\frac{1}{2}}\Big)
\end{eqnarray}
which agrees with the refined topological string partition function in \cite{IKV}. The second way of gluing the new vertices gives the same refined partition function:

\begin{eqnarray}\displaystyle
Z&=&\sum_{\lambda}(-Q)^{|\lambda|}\,T_{\lambda\,\emptyset\,\emptyset}(t,q)\,T_{\lambda^t\,\emptyset\,\emptyset}(q,t)\\\nonumber
&=&\sum_{\sigma_{1}\,\sigma_{2}}\,(-Q)^{|\sigma_{1}|}\,f_{\sigma_{1}}^{2}(t,q)\,f_{\sigma_{2}}^{2}(q,t)\,P_{\sigma_{1}}(t^{\rho};q,t)\,P_{\sigma_{2}}(q^{\rho};t,q)\,\sum_{\lambda}U_{\lambda^{t}\,\sigma_{1}}(t,q)\,U_{\lambda\,\sigma_{2}}(q,t)\\\nonumber
&=&\sum_{\sigma_{1}}(-Q)^{|\sigma_{1}|}\,P_{\sigma_{1}}(t^{\rho};q,t)P_{\sigma_{1}^t}(q^{\rho};t,q)=\prod_{i,j}\Big(1-Q\,q^{i-\frac{1}{2}}\,t^{j-\frac{1}{2}}\Big)
\end{eqnarray}
where in simplifying the second line we have used
\begin{eqnarray}\displaystyle
f_{\eta}(t,q)\,f_{\eta^t}(q,t)=1\,,\,\,\,\,\,\,\,U^{T}(t,q)\,J\,U(q,t)=J\,\,\,\,\,\mbox{where}\,\,\,\,\,(J)_{\sigma\,\eta}=\delta_{\sigma\,\eta^t}
\end{eqnarray}
\subsection{Example 2: Local ${\bf P}^{1}\times{\bf P}^{1}$}
\par{The topological string partition of local ${\bf P}^{1}\times{\bf  P}^{1}$ can be computed using the refined topological vertex \cite{IKV}. Using the new topological vertex we can re-calculate its partition function but with the different choice of the preferred direction which was not allowed according to the gluing rules of the refined topological vertex. \figref{localp1p1} demonstrates the basic idea and the choice of gluing.  }

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{p1p1.eps}
\end{center}
\caption{The computation of the partition function of local ${\bf P}^{1}\times{\bf P}^{1}$ using the new topological vertex. Circled vertices are the ones where the new vertex is used.}\label{localp1p1}
\end{figure}

From our previous discussion of the new topological vertex we can rewrite this expression as
\begin{eqnarray}\displaystyle\nonumber
Z=\sum_{\lambda\,\mu\, \delta\, \sigma}(-Q_{b})^{|\lambda|+|\delta|}(-Q_{f})^{|\mu|+|\sigma|}&&\widetilde{f}_{\lambda}(t,q)\widetilde{f}_{\mu}(t,q)\widetilde{f}_{\sigma}(q,t)
\widetilde{f}_{\delta}(q,t)\,C_{\lambda\,\mu^t\,\emptyset}(q,t)
\,C_{\delta\,\sigma^t\,\emptyset}(t,q)\\\nonumber
&&\times\,T_{\sigma\,\lambda^t\,\emptyset}(t,q)\,T_{\mu\,\delta^t\,\emptyset}(q,t)
\end{eqnarray}

\subsection{Example 3: Local ${\bf F_{1}}$}

The local ${\bf F}_{1}$ refined partition was calculated in section 3 for a particular choice of the preferred direction. Here we use the new vertex to calculate it using a different preferred direction. The web diagram of local ${\bf F}_{1}$ and the choice of the preferred direction is shown in \figref{f1}.

\begin{figure}[h]
  \centering

  \includegraphics[width=1.65in]{f1.eps}\\
  \caption{The choice of preferred edges for local ${\bf F}_{1}$.}\label{f1}
\end{figure}

The partition function can be calculated easily using the rules discussed in the last section and is given by
\begin{eqnarray}\displaystyle
Z(Q_{b},Q_{f})&=&\sum_{\lambda\,\mu\,\tau\,\nu}(-Q_{b})^{|\nu|+|\tau|}Q_{f}^{|\lambda|+|\mu|+|\tau|}
f_{\tau}(q,t)^2\,f_{\lambda}(t,q)f_{\mu}(q,t)\\\nonumber
&&\,C_{\lambda\nu \emptyset}(t,q)C_{\emptyset\nu^t\mu}(q,t)C_{\tau\emptyset\mu^t}(t,q)T_{\tau^t\lambda^t\emptyset}(t,q)
\end{eqnarray}

\subsection{Example 4}

Consider the geometry shown in \figref{twop2}. In this we have two 4-cycles both of them $\mathbf{P}^2$ joined by a $\mathbf{P}^1$. There are three K\"{a}hler parameters $(t_{H_{1}},t_{H_{2}},t_{E})$  corresponding to the curves $H_{1}$ (the hyperplane class of $\mathbf{P}^2$), $H_{2}$ (the hyperplane class of the second $\mathbf{P}^2$) and the exceptional curve $E$,
\begin{eqnarray}\displaystyle
Q_{H_{i}}=\mbox{exp}(-t_{H_{i}})\,,\,\,\,\,Q=\mbox{exp}(-t_{E})\,.
\end{eqnarray}

\begin{figure}[h]
  \centering

  \includegraphics[width=1.75in]{twop2.eps}\\
  \caption{The choice of preferred edges for the geometry with 2 4-cycles.}\label{twop2}
\end{figure}
The partition function of this geometry can be easily determine using the refined vertex and the new vertex. The two horizontal lines we take to be the preferred edges of the refined vertex. Then the refined partition function of this geometry is given by
\begin{eqnarray}\displaystyle\nonumber
Z(Q_{H_{1}},Q_{H_{2}},Q)&=&\sum_{\lambda\,\mu\,\nu\,\sigma\,\eta\,\tau}(-Q_{H_{1}})^{|\lambda|+|\mu|+|\nu|}(-Q_{H_{2}})^{|\sigma|+|\eta|+|\tau|}(-Q)^{|\zeta|}\\\nonumber
&&
(f_{\nu}(q,t)f_{\lambda}(t,q)f_{\mu}(q,t)f_{\tau}(t,q)f_{\sigma}(q,t)f_{\eta}(t,q))^2\\\nonumber
&&\,C_{\lambda\emptyset\nu}(t,q)C_{\emptyset\mu\nu^t}(q,t)C_{\emptyset\,\sigma^t\tau}(t,q)C_{\eta^t\emptyset\,\tau^t}(q,t)T_{\lambda^t\mu^t\zeta}(t,q)T_{\sigma^t\eta^t\zeta^t}(q,t)\,.
\end{eqnarray}

\section{Relation with refined Chern-Simons theory}

In this section, we will clarify the how the refined Chern-Simons theory \cite{AS} is related to the refined vertex and the new vertex.

The refined Chern-Simons theory is formulated as the supersymmetric index of M-theory compactified on $(X\times\mathbb{C}^2)\times_{t,q}S^{1}$ with $N$ M5-branes on $(L\times \mathbb{C})\times_{t,q}S^{1}$, where the fibered product means that
 \begin{eqnarray}\displaystyle
 \theta\in S^{1}\mapsto \theta+2\pi \implies (z_{1},z_{2})\mapsto (q\,z_{1},t^{-1}z_{2})\,,\,\,X\mapsto w(X)
 \end{eqnarray}
 where $w$ is a $U(1)\times U(1)$ action on $X$ which preserves the holomorphic 3-form and $L$ is a Lagrangian 3-cycle. When $X=T^{*}S^3$ and $L=S^{3}$ the large $N$ transition takes the above geometry to resolved coinfold \cite{Gopakumar:1998ki,AS}.\\

\noindent Consider $T^{*}S^3$ with $N$ M5-branes wrapped on $S^{3}$ and extending in the $z_{2}$-plane along with two stacks of M5-branes wrapped on the Lagrangian cycles ${\cal L}_{1}$ and ${\cal L}_{2}$, which have the topology of $S^{1}\times \mathbb{R}^{2}$, and extending in the $z_{2}$-plane. In the $S^{3}$ the boundary of these Lagrangian cycles are the two components of the Hopf link. The Lagrangian cycles are shown as dotted line in \figref{flop2}. Compactifying this brane configuration on  circle to go to type IIA string theory we get $N$ D4-branes wrapped on the $S^{3}$ with D2-branes ending on these D4-branes giving the open string instantons \cite{OV}. Now going through the large $N$ transition the singular geometry, when the $S^{3}$ has shrunk, can resolve itself in two different ways related to each other by a flop as shown in \figref{flop2}.

\begin{figure}[h]
  \centering

  \includegraphics[width=3in]{flop2.eps}\\
  \caption{The large $N$ transition can resolve the conifold in two different ways. Resolution given in (a) leads to refined Chern-Simons S-matrix $P_{\lambda}(t^{-\rho};q,t)P_{\mu}(t^{-\rho}q^{-\lambda};q,t)$ for large $N$. The resolution of type (b) leads to the refined Hopf link calculated using the refined topological vertex.}\label{flop2}
\end{figure}

The normalized open string partition function of the geometry before the transition is then given by \cite{AS}
\begin{eqnarray}\displaystyle
Z(U_{1},U_{2})&=&\sum_{\lambda\,\mu}Z_{\lambda\,\mu}\,P_{\lambda}(U_{1};q,t)\,P_{\mu}(U_{2};q,t)\\\nonumber
Z_{\lambda\,\mu}&=&P_{\lambda}(t^{-\rho_{N}};q,t)P_{\mu}(t^{-\rho_{N}}q^{-\lambda};q,t)\,,
\end{eqnarray}
where $U_{1}$ and $U_{2}$ are the holonomies associated with the two branes, $\rho_{N}=\{-\frac{1}{2},-\frac{3}{2},\cdots,-N+\frac{1}{2}\}$ and $\lambda,\mu$ are representations of $SU(N)$. It was shown in \cite{IK} that $Z(U_{1},U_{2})$ is also the partition function of the resolved conifold with branes but taking the K\"ahler parameter of the resolved conifold $Q=t^{-N}$ so that $N\mapsto \infty$ corresponds $Q\mapsto \infty$. This implies that when large $N\mapsto +\infty$ transition takes the brane configuration to \figref{flop2}(a) then the large $N\mapsto -\infty$ takes us to \figref{flop2}(b). \figref{flop2}(a) gives the refined Chern-Simons Hopf link invariant, $P_{\lambda}(t^{-\rho_{N}};q,t)P_{\mu}(t^{-\rho_{N}}q^{-\lambda};q,t)$,  and the \figref{flop2}(b) gives the refined topological vertex refined Hopf link.

However we can also consider a configuration of branes in which one stack of D4-branes fill the $z_{1}$-plane and the other fills the $z_{2}$-plane as shown in \figref{flop3}. In this case the large $N$ transition $N\mapsto +\infty$ shown in \figref{flop3}(a) gives the refined topological vertex expressed in the basis of Macdonald polynomials.

\begin{figure}[h]
  \centering

  \includegraphics[width=3in]{flop3.eps}\\
  \caption{The large $N$ transition with a different configuration of branes.}\label{flop3}
\end{figure}

.
\subsection{Local $\mathbf{P}^{2}$ from refined Chern-Simons Theory}

The local $\mathbb{P}^{2}$ partition function we obtained in section 3 can also be obtained using the refined Chern-Simons theory\footnote{See \cite{AS2} for related work.}.

In section 3 we showed that slicing of the local $\mathbb{P}^2$ web diagram by $\epsilon_{\alpha}$ branes required a vertex which has $\epsilon_{1}$ branes on both legs. We obtained this vertex by flop transition from the refined Hopf link. This two-legged vertex is the same as the refined Hopf link S-matrix (in the basis of Schur functions). Thus we can obtain the local $\mathbb{P}^{2}$ partition function blowing one of the vertices, shrinking the $\mathbb{P}^{1}$, deforming it to $S^{3}$ and putting refined Chern-Simons theory on the $S^{3}$ as shown in
\figref{rcs1}.

\begin{figure}[h]
  \centering

  \includegraphics[width=3in]{rcs.eps}\\
  \caption{The refined partition function of the blowup of local ${\bf P}^{2}$ can be calculated by putting refined Chern-Simons theory on the $S^{3}$. The refined topological vertex is associated with the other two vertices of the geometry. The large $N$ limit then gives the refined partition function of the local ${\bf P}^{2}$.}\label{rcs1}
\end{figure}

The local $\mathbb{P}^{2}$ web diagram can also be sliced in a different way as well. This is shown in \figref{rcspp}. In this case we see that vertices are of the $(\epsilon_{1},\epsilon_{1})$, $(\epsilon_{2},\epsilon_{2})$ and $(\epsilon_{1},\epsilon_{2})$ type\footnote{By a vertex of $(\epsilon_{\alpha},\epsilon_{\beta})$ type we mean the corresponding branes fill the $z_{\alpha}$-plane and the $z_{\beta}$-plane.}.

\begin{figure}[h]
\centering

  \includegraphics[width=4in]{rcsp2.eps}\\
  \caption{The refined partition function of local ${\bf P}^{2}$ can be obtained from just the refined Chern-Simons theory as well by wrapping appropriate branes on the $S^{3}$'s and the taking the large $N$ limit.}\label{rcspp}
\end{figure}

 Thus we can write down a different representation of the refined partition function of local $\mathbb{P}^{2}$:
\begin{eqnarray}\displaystyle\label{csss}
Z_{local\,\mathbb{P}^2}(Q,t,q)&=&\sum_{\nu\,\alpha\,\beta}(-Q)^{|\lambda|+|\mu|+|\nu|}\Big(\frac{t}{q}\Big)^{\frac{|\lambda|-|\mu|}{2}}\,
f_{\lambda}(t,q)f_{\mu}(t,q)f_{\nu}^{2}(q,t)\\\nonumber
&&\Big(P_{\lambda}(t^{-\rho};q,t)\,P_{\mu}(t^{-\rho}\,q^{-\lambda};q,t)\Big)
\Big(P_{\nu}(q^{-\rho};t,q)P_{\lambda^t}(q^{-\rho}t^{-\nu})\Big)\\\nonumber
&&\Big[P_{\nu^t}(t^{-\rho};q,t)P_{\mu^t}(q^{-\rho}t^{-\nu};t,q)\Big]
\end{eqnarray}

The first two factors in second line of the above equation are S-matrices for the refined Chern-Simons theory in which for one of factors  $t\leftrightarrow q$. The last factor is different from the first two. The last factor is the refined topological vertex in the basis of Macdonald functions.

\section{Conclusions}
We have defined a new vertex which can be used along with the refined vertex to calculate the refined partition function of any non-compact toric Calabi-Yau threefold. We believe that this new vertex is closely related to the ``index vertex" of Okounkov and Nekrasov \cite{ONtalk}. We also showed that refined amplitudes from refined Chern-Simons theory for branes in different $z_{\alpha}$-planes can also be calculated from the refined vertex formalism extended by the new vertex.

\noindent\section*{ Acknowledgments}
We would like to thank Mina Aganagic, Johannes Walcher and Cumrun Vafa for many useful discussions. We would also like to thank the Simon Center for Geometry and Physics for hospitality during the Simons workshop 2011 where part of this work was carried out. AI was partly supported by the  Higher Education Commission grant HEC-2052. CK was partly supported by the INFN Research Project TV12.

\section{Appendix A}

In this appendix, we sketch the proof that the partition function of the local ${\bf P}^2$ computed from the refined vertex Eq. (\ref{part}) and the one from the refined Chern-Simons theory Eq. (\ref{csss}) are equal.  The section is rather technical and we will try to included all the necessary identities that are used in the manipulations. Let us remind the reader the refined partition function of local $\mathbf{P}^{2}$:
\begin{align}\label{part}
Z_{local\,\mathbf{P}^{2}}&=\sum_{\substack{\lambda\,\mu\,\nu\\\eta\,\sigma}} (-Q_{H})^{|\lambda|+|\mu|+|\nu|}(-1)^{|\lambda|+|\mu|}\left(\frac{t}{q} \right)^{|\lambda|-|\mu|}f_{\nu}^{2}(q,t)N^{\eta}_{\lambda\mu}U_{\eta\sigma}f_{\sigma}P_{\sigma}(t^{-\rho};q,t)\\\nonumber &\times P_{\nu^{t}}(t^{-\rho};q,t)P_{\nu}(q^{-\rho};t,q)s_{\lambda}(t^{-\nu}q^{-\rho})s_{\mu}(q^{-\rho}t^{-\nu}).
\end{align}
First, we want to use the following identity to change the Littlewood-Richardson coefficient to its two parameter extension
\begin{eqnarray}\displaystyle
\sum_{\alpha\,\beta}\widehat{N}^{\sigma}_{\alpha\,\beta}\,U_{\lambda\,\alpha}U_{\mu\,\beta}=\sum_{\eta}N^{\eta}_{\lambda\,\mu}U_{\eta\,\sigma}
\end{eqnarray}
The reason behind this change becomes obvious when the following non-trivial identity is recalled\footnote{In \cite{IK}, this identity played a very crucial role. The definition of the framing factors $f_{\lambda}$ differs by a factor of $(-1)^{|\lambda|}$ there, but note that $\widehat{N}_{\lambda\,\mu}^{\nu}$ vanish unless $|\nu|=|\lambda|+|\mu|$, \textit{i.e.}, the identity is also valid when $f$'s are replaced with respect to our current definition of framing factors.  }
\begin{eqnarray}\displaystyle\label{id1}
P_{\lambda}(t^{-\rho};q,t)P_{\mu}(t^{-\rho}q^{-\lambda};q,t)=\frac{1}{f_{\lambda}(t,q)f_{\mu}(t,q)}\sum_{\nu}\widehat{N}_{\lambda\,\mu}^{\nu}f_{\nu}(t,q)P_{\nu}(t^{-\rho};q,t).
\end{eqnarray}
After using this identity, we will have to perform sums of the following form\footnote{In this section, for brevity of notation, we will use $U_{\lambda\,\mu}$ to mean $U_{\lambda\,\mu}$(q,t) and $\widetilde{U}_{\lambda\,\mu}$ for $\widetilde{U}_{\lambda\,\mu}(t,q)$.}
\begin{eqnarray}\displaystyle
\sum_{\lambda}U_{\lambda\,\alpha}s_{\lambda}(\gamma\,t^{-\nu}q^{-\rho}),
\end{eqnarray}
where $\gamma$ is just a factor. This part gets a little bit more involved, we will need to recall some facts of symmetric functions. There exists an involution which acts on the ring of symmetric functions. In particular, it takes the power sum symmetric function to a multiple of them:
\begin{eqnarray}\displaystyle
\omega\cdot p_{n}=(-1)^{n-1}p_{n}.
\end{eqnarray}
One can show that the action of this involution on the Schur functions is less trivial:
\begin{eqnarray}\displaystyle
\omega\cdot s_{\lambda}=s_{\lambda^{t}},
\end{eqnarray}
for all partitions $\lambda$. This involution has a two parameter extension $\omega_{t,q}$ which we will use in the following manipulations. Its action on the power sum symmetric function is given by
\begin{eqnarray}\displaystyle
\omega_{t,q}\cdot p_{n}=(-1)^{n-1}\frac{1-t^{n}}{1-q^{n}}p_{n},
\end{eqnarray}
for $q,t\neq 1$. In \cite{Taki:2008hb}, Taki derived an identity using the involution which can be massaged to
\begin{eqnarray}\displaystyle
s_{\lambda}(\gamma\, q^{-\rho}t^{-\nu})=\omega_{t,q}\cdot s_{\lambda^{t}}(-\gamma\,t^{\rho}q^{\nu^{t}}).
\end{eqnarray}
Using this we can write
\begin{align}
s_{\lambda}(x)&=\omega_{t,q}\cdot s_{\lambda^{t}}(x^{*})\\ \nonumber
&=\omega_{t,q}\cdot\sum_{\eta}\widetilde{U}_{\lambda^{t}\,\eta}\,P_{\eta}(x^{*};t,q)\\\nonumber
&=\sum_{\eta}\widetilde{U}_{\lambda^{t}\,\eta}\,\omega_{t,q}\cdot P_{\eta}(x^{*};t,q)\\\nonumber
&\equiv\sum_{\eta}\widetilde{U}_{\lambda^{t}\,\eta}\, \widehat{P}_{\eta}(x^{*};t,q)
\end{align}
where we defined  $x=Q_{H}t^{1/2}q^{-1/2}\,q^{-\rho}t^{-\nu}$ and $x^{*}=-Q_{H}t^{1/2}q^{-1/2}\,t^{\rho}q^{\nu^{t}}$. $\widehat{P}_{\eta}(x^{*};t,q)$ denotes the action of $\omega_{t,q}$ on the Macdonald function. The involution $\omega_{t,q}$ passed through the sum since it acts on the symmetric functions, rather that $q,t$-dependent factors. Now we are ready to perform the above mentioned sum over $\lambda$:
\begin{align}\label{id2}
\sum_{\lambda}U_{\lambda\,\alpha} s_{\lambda}(x)&=\sum_{\lambda\,\eta}U_{\lambda\,\alpha}\widetilde{U}_{\lambda^{t}\,\eta}\widehat{P}_{\eta}(x^{*},t,q)\\\nonumber
&=\sum_{\eta}\left(\sum_{\lambda}\left(\widetilde{U}^{-1} \right)_{\alpha^{t}\,\lambda^{t}}\widetilde{U}_{\lambda^{t}\,\eta} \right)\widehat{P}_{\eta}(x^{*};t,q)\\\nonumber
&=\widehat{P}_{\alpha^{t}}(x^{*};t,q),
\end{align}
where we have used
\begin{eqnarray}\displaystyle
U^{T}JU=J\implies (U^{-1})_{\alpha\,\beta}=\widetilde{U}_{\beta^t\,\alpha^t}.
\end{eqnarray}
We need to know the action of $\omega_{t,q}$ on the Macdonald function which is given in \cite{macdonald} to be
\begin{eqnarray}\displaystyle
\omega_{t,q}\cdot P_{\alpha^{t}}(x^{*};t,q)=Q_{\alpha}(x^{*};q,t).
\end{eqnarray}
The desired form is obtained by using the following form of the identity Eq. B.29 from \cite{Awata:2005fa}
\begin{eqnarray}\displaystyle
Q_{\lambda}(q^{\nu^{t}}t^{\rho};q,t)=(-1)^{|\lambda|}\left(\frac{t}{q}\right)^{|\lambda|/2}\,P_{\lambda^{t}}(t^{-\nu}q^{-\rho};t,q).
\end{eqnarray}
Using Eq. (\ref{id1}) and Eq. (\ref{id2}), we can easily show that Eq. (\ref{part}) is equal to Eq. (\ref{csss}).

\section{Appendix B}
In this section, we want to review our notations, conventions and collect some of the useful formulas for the derivations. We want to warn the reader that the convention we follow in this note is different from the one used in \cite{IKV}. The difference comes from the pictorial representation of the Young diagrams. In  \cite{IKV}, for a given representation $\lambda=\{\lambda_{1},\lambda_{2},\mathellipsis \}$, we illustrated the corresponding Young diagram with columns of height $\lambda_{i}$. The arm length $a(i,j)$ and leg length $\ell(i,j)$ of a given box $(i,j)$ in the diagram were defined as the number of boxes to the right of that box and above that box, respectively. In this note we follow the more conventional notation shown in \figref{convention} where $\lambda_{i}$'s are the number of boxes in the $i^{th}$ row. The arm and leg lengths are defined accordingly.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{convention.eps}
\end{center}
\caption{The figure shows the present convention. $\lambda=\{5,3,3,2,1,1\}$}\label{convention}
\end{figure}
In addition to the arm and leg lengths, we define the co-arm length $a'(i,j)$ and co-leg length $\ell'(i,j)$
\begin{align}
&a(i,j)=\nu_{i}-j\,,\,\,\,\,\ell(i,j)=\nu^{t}_{j}-i\\
&a'(i,j)=j-1\,,\,\,\,\ell'(i,j)=i-1
\end{align}
Note that
\begin{eqnarray}\displaystyle
a_{\lambda}(i,j)=\ell_{\lambda^{t}}(j,i),\qquad\qquad\ell_{\lambda}(i,j)=a_{\lambda^{t}}(j,i)
\end{eqnarray}
with $(i,j)\in\lambda$ and $(j,i)\in\lambda^{t}$.
The difference in the conventions reflects also on some identities used in \cite{IKV}, here we use
\begin{align}
&n(\lambda)=\sum_{i=1}^{\ell(\lambda)}(i-1)\lambda_{i}=\frac{1}{2}\sum_{i=1}^{\ell(\lambda)}\lambda^{t}_{i}(\lambda^{t}_{i}-1)=\sum_{(i,j)\in\lambda}\ell(i,j)=\sum_{(i,j)\in\lambda}\ell'(i,j)=\frac{\Arrowvert\lambda^{t}\Arrowvert^{2}}{2}-\frac{\lambda}{2}\\
&n(\lambda^{t})=\sum_{i=1}^{\ell(\lambda^{t})}(i-1)\lambda^{t}_{i}=\frac{1}{2}\sum_{i=1}^{\ell(\lambda^{t})}\lambda_{i}(\lambda_{i}-1)=\sum_{(i,j)\in\lambda}a(i,j)=\sum_{(i,j)\in\lambda}a'(i,j)=\frac{\Arrowvert\lambda\Arrowvert^{2}}{2}-\frac{\lambda}{2}
\end{align}
with $\ell(\lambda)$ being the number of non-zero $\lambda_{i}$'s, or in other words, it is the number of rows in the Young diagram. The hook length $h(i,j)$ and the content $c(i,j)$ are defined as
\begin{eqnarray}\displaystyle
h(i,j)=a(i,j)+\ell(i,j)+1,\qquad c(i,j)=j-i
\end{eqnarray}
which satisfy
\begin{align}
&\sum_{(i,j)\in\lambda}h(i,j)=n(\lambda^{t})+n(\lambda)+|\lambda|,\\
&\sum_{(i,j)\in\lambda}c(i,j)=n(\lambda^{t})-n(\lambda).
\end{align}
The refined topological vertex in this convention is given by
\begin{eqnarray}\displaystyle\nonumber
C_{\lambda\,\mu\,\nu}(t,q)&=&\Big(\frac{q}{t}\Big)^{\frac{||\mu||^2}{2}}\,t^{\frac{\kappa(\mu)}{2}}\,q^{\frac{||\nu||^2}{2}}\,\widetilde{Z}_{\nu}(t,q)
\sum_{\eta}\Big(\frac{q}{t}\Big)^{\frac{|\eta|+|\lambda|-|\mu|}{2}}\,s_{\lambda^{t}/\eta}(t^{-\rho}\,q^{-\nu})\,s_{\mu/\eta}(t^{-\nu^t}\,q^{-\rho})\\
\end{eqnarray}
where $s_{\lambda/\eta}(\mathbf{x})$ is the skew-Schur function and $\widetilde{Z}_{\nu}(t,q)$ is given by
\begin{eqnarray}\displaystyle
\widetilde{Z}_{\nu}(t,q)&=&\prod_{(i,j)\in \nu}\Big(1-q^{a(i,j)}\,t^{\ell(i,j)+1}\Big)^{-1},
\end{eqnarray}
which is related to the Macdonald polynomials
\begin{eqnarray}\displaystyle
t^{\frac{\Arrowvert\nu^t\Arrowvert^2}{2}}\,\widetilde{Z}_{\nu}(t,q)&=&P_{\nu}(t^{-\rho};q,t),\\
q^{\frac{\Arrowvert\nu\Arrowvert^2}{2}}\,\widetilde{Z}_{\nu^t}(q,t)&=&P_{\nu^t}(q^{-\rho};t,q).
\end{eqnarray}
For the Macdonald polynomials with this special set of argument we have
\begin{eqnarray}\displaystyle
P_{\lambda}(t^{\rho};q,t)=(-1)^{|\lambda|}q^{n(\lambda^{t})}t^{-n(\lambda)}P_{\lambda}(t^{-\rho};q,t).
\end{eqnarray}
The Macdonald polynomials also satisfy
\begin{eqnarray}\displaystyle
P_{\lambda}(\mathbf{x};q,t)=P_{\lambda}(\mathbf{x};q^{-1},t^{-1})
\end{eqnarray}
For completeness, let us also review some identities regarding to Schur functions and MacDonald polynomials:
\begin{align}
&s_{\lambda}(\mathbf{x})s_{\mu}(\mathbf{x})=\sum_{\eta}N_{\lambda\mu}^{\eta}s_{\eta}(\mathbf{x})\label{schurfactor}\\
&s_{\lambda/\mu}(\mathbf{x})=\sum_{\eta}N^{\lambda}_{\mu\eta}s_{\eta}(\mathbf{x})\\
&P_{\lambda}(\mathbf{x};q,t)P_{\mu}(\mathbf{x};q,t)=\sum_{\eta}\widehat{N}^{\eta}_{\lambda\mu}P_{\eta}(\mathbf{x};q,t)\label{macdonald}\\
&s_{\eta}(\mathbf{x})=\sum_{\sigma}U_{\eta\sigma}P_{\sigma}(\mathbf{x};q,t)\label{schurmacdonald}
\end{align}
The following relations among Macdonald polynomials prove to be very useful for our computations
\begin{eqnarray}\displaystyle\label{ratio}
\frac{P_{\mu}(q^{-\lambda}\,t^{-\rho};q,t)}{P_{\mu}(t^{-\rho};q,t)}&=&\frac{P_{\lambda}(q^{-\mu}\,t^{-\rho};q,t)}{P_{\lambda}(t^{-\rho};q,t)}\\
P_{\sigma}(t^{-\rho},z\,t^{\rho};q,t)&=&P_{\sigma}(t^{-\rho};q,t)\prod_{s\in \sigma}(1-z\,q^{a'(s)}\,t^{-\ell'(s)})\label{product}
\end{eqnarray}
The following sum rules are essential for vertex computations
\begin{align}
&\sum_{\eta}s_{\eta/\lambda}(\mathbf{x})s_{\eta/\mu}(\mathbf{y})=\prod_{i,j=1}^{\infty}(1-x_{i}y_{j})^{-1}\sum_{\tau}s_{\mu/\tau}(\mathbf{x})s_{\lambda/\tau}(\mathbf{y})\\
&\sum_{\eta}s_{\eta^{t}/\lambda}(\mathbf{x})s_{\eta/\mu}(\mathbf{y})=\prod_{i,j=1}^{\infty}(1+x_{i}y_{j})\sum_{\tau}s_{\mu^{t}/\tau}(\mathbf{x})s_{\lambda^{t}/\tau^{t}}(\mathbf{y})\\
&\sum_{\eta}P_{\eta}(\mathbf{x};q,t)P_{\eta^{t}}(\mathbf{x};t,q)=\prod_{i,j=1}^{\infty}(1+x_{i}y_{j})
\end{align}
We have considered the normalized amplitudes, both the open and closed amplitudes are infinite series in the K\"{a}hler parameters, however, their ratio is finite as a result of the following identity
\begin{eqnarray}\displaystyle\label{infinite}
\prod_{i,j=1}^{\infty}\frac{1-Q\,q^{-\lambda_{i}+j-1/2}t^{i-1/2}}{1-Q\,q^{j-1/2}t^{i-1/2}}=\prod_{s\in\lambda}\left (1-Q\sqrt{\frac{t}{q}}\, q^{-a'(s)}t^{\ell(s)}\right).
\end{eqnarray}

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Q: How do PHP superglobals like $_POST get populated by an HTML form? I understand if I create an html form with the action attribute set to a PHP script, I can access the POST data via the super global $_POST. My question is how does this happen?
If I have a simple form like this:
<form action="script.php" method="POST">
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  <input type="submit" value="Submit">
</form>

And a simple script like this:
<?php
    echo $_POST["name"];
?>

What mechanism brings the raw form data from the POST request into PHP? Is there some sort of CGI Script? What would the equivalent be for another language, say Ruby?

A:
An associative array of variables passed to the current script via the
  HTTP POST method when using application/x-www-form-urlencoded or
  multipart/form-data as the HTTP Content-Type in the request.
$HTTP_POST_VARS contains the same initial information, but is not a
  superglobal. (Note that $HTTP_POST_VARS and $_POST are different
  variables and that PHP handles them as such)

https://www.php.net/manual/en/reserved.variables.post.php
It's just a super global variable that the PHP interpreter stores all post data in until the next request on which it is cleared.

A: When you submit a form through the browser, the browser will take the data and serialize it, which then gets send to the web server in a HTTP/HTTPS request.
For GET forms the data is serialized and appended to string query in your URL e.g.myphp.php?field1=foo&field2=bar
For POST data it is inserted in the body of the request.
Based on the type of the request PHP engine will then parse the data received and for your convenience url decode it and populate the superglobals with it. In the example above the data will be like this:
$_GET = [
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];

This is a nice feature of PHP in contrast to language like PERL which force you to parse the input data yourself.
Additionally PHP offers a way to read the input stream directly for the raw data which you would normally receive.

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