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A Geometric Approach to the Links-Quivers Correspondence II: Rational Links

Jonathan A. Higgins

Abstract

The Links-Quivers Correspondence predicts that the generating function for the symmetric (or antisymmetric) colored HOMFLY-PT polynomials for links can be put in a "quiver form," so that the generating function is expressed in terms of a quadratic form and two linear forms. This was originally proved for rational links by Stosic and Wedrich, but here we give a direct geometric description of the linear and quadratic forms in terms of the first and second configuration spaces of the 3-punctured plane.

A Geometric Approach to the Links-Quivers Correspondence II: Rational Links

Abstract

The Links-Quivers Correspondence predicts that the generating function for the symmetric (or antisymmetric) colored HOMFLY-PT polynomials for links can be put in a "quiver form," so that the generating function is expressed in terms of a quadratic form and two linear forms. This was originally proved for rational links by Stosic and Wedrich, but here we give a direct geometric description of the linear and quadratic forms in terms of the first and second configuration spaces of the 3-punctured plane.
Paper Structure (28 sections, 34 theorems, 145 equations, 28 figures)

This paper contains 28 sections, 34 theorems, 145 equations, 28 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum Algebra
  4. Rational Tangles
  5. Closures of Rational Tangles
  6. Colored HOMFLY-PT Polynomials of Rational Links
  7. The Links-Quivers Correspondence
  8. Links-Quivers for Rational Tangles
  9. Construction of Lagrangians
  10. Loops and Homomorphisms
  11. Geometric Formulas for Rational Tangles
  12. Rational Knots
  13. Setting Up the Theorem for Knots
  14. Symmetry of $Q$
  15. Step 1: The Linear Forms
  16. ...and 13 more sections

Key Result

Theorem 1

For a rational knot $K_{u/v}$, we may express $P(K_{u/v})$ as where $S$ and $A$ are computed by winding numbers of loops based at the $u$ Lagrangian intersections in $\mathcal{D}(K_{u/v})$ about the three punctures. The quadratic form $Q$ may be computed similarly, but by passing to the second configuration space of the 3-punctured plane, where we also need to

Figures (28)

  • Figure 1: Left: $\mathcal{D}(K_{5/2})$. Right: $\mathcal{D}(L_{8/3}).$
  • Figure 2: Left: the trivial tangle, $\tau_{0/1}$. Center: top twist rule. Right: right twist rule.
  • Figure 3: The figure-8 knot, $K_{5/2}$, taken as $\text{Cl}(\tau_{5/2})$
  • Figure 4: The three types of basis webs
  • Figure 5: An application of Lemma \ref{['bridgerecipe']} to $\tau_{4/3}$
  • ...and 23 more figures

Theorems & Definitions (90)

  • Conjecture 1: Links-Quivers Correspondence
  • Theorem 1
  • Theorem 2
  • Conjecture 2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • ...and 80 more