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A Geometric Approach to the Links-Quivers Correspondence I: Rational Tangles

Jonathan A. Higgins

Abstract

The Links-Quivers Correspondence predicts that all the symmetric (or antisymmetric) colored HOMFLY-PT polynomials of a link can be recovered from a finite amount of data (a quiver) associated to the link. We give a new geometric proof of the Links-Quivers Correspondence modified for rational tangles and explicitly describe the corresponding quivers in terms of winding numbers in the punctured plane and its second configuration space.

A Geometric Approach to the Links-Quivers Correspondence I: Rational Tangles

Abstract

The Links-Quivers Correspondence predicts that all the symmetric (or antisymmetric) colored HOMFLY-PT polynomials of a link can be recovered from a finite amount of data (a quiver) associated to the link. We give a new geometric proof of the Links-Quivers Correspondence modified for rational tangles and explicitly describe the corresponding quivers in terms of winding numbers in the punctured plane and its second configuration space.
Paper Structure (22 sections, 23 theorems, 130 equations, 22 figures)

This paper contains 22 sections, 23 theorems, 130 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Some Quantum Algebra
  4. Rational Tangles
  5. $\langle-\rangle_j$ and $\langle\langle-\rangle\rangle_j$ for rational tangles
  6. Prior Results for Computing $\langle\langle\tau_{u/v}\rangle\rangle_j$
  7. Links-Quivers for Rational Tangles
  8. The Main Theorem
  9. Setting Up the Theorem for Poincaré Polynomials
  10. Symmetry of $Q$
  11. Specializing to the HOMFLY-PT polynomial
  12. Proof of Main Theorem
  13. Some More Loops in $\text{Conf}^2(M)$
  14. Proof of Case 1
  15. Proof of Case 2
  16. ...and 7 more sections

Key Result

Theorem 1

Given a rational tangle $\tau_{u/v}$, $\mathcal{P}(\tau_{u/v})$ may be written as where $S,A,$ and $T$ may be computed via winding numbers about the three punctures in $\mathcal{D}(\tau_{u/v})$ of loops based at the Lagrangian intersections. Furthermore, $Q$ may be computed in a similar manner by passing to the second configuration space of the 3-punctured plane, where we also ne

Figures (22)

  • Figure 1: The diagram $\mathcal{D}(\tau_{5/2})$ used in Theorem \ref{['introtanglethm']} for the tangle $\tau_{5/2}$
  • Figure 2: Left: the trivial tangle $\tau_{0/1}$. Center: top twist rule. Right: right twist rule. For twist rules, $\tau$ denotes the tangle being twisted, and the orientation is determined by the orientation of endpoints on $\tau$.
  • Figure 3: Left: the four colored arcs connecting the branch points on the boundary $S^2$. Right: the disk separating the trivial tangle in $B^3$.
  • Figure 4: An application of Lemma \ref{['bridgerecipe']} to $\tau_{4/3}$ (top) and $\tau_{3/4}$ (bottom)
  • Figure 5: Examples of the $\mathcal{D}(\tau_{u/v})$ notation
  • ...and 17 more figures

Theorems & Definitions (70)

  • Theorem 1
  • Conjecture 1.1: Links-Quivers Correspondence
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3: Proposition 1.7, AM20
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 60 more