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Knot contact homology as a planar limit of Chern-Simons theory

Ben Webster, Meri Zaimi

TL;DR

The paper develops a skein-theoretic, Deligne-style planar-limit framework to connect the augmentation varieties of knots and links with the large-$N$ limit of Chern-Simons theory. It introduces the HOMFLYPT difference module as a module over a quantum torus, built from heavy and light objects in a planar limit of the HOMFLYPT spider category, and shows that its degree-0 classical limit recovers abelianized knot contact homology (KCH). The central result is an isomorphism between KCH and the planar-limit invariant $\mathsf{H}(K)$, with a conjectural identification of the augmentation ideal as the classical limit of the quantum-parameterized recursion ideal for colored HOMFLYPT polynomials, extending to links. Together these contributions provide an algebraic, skein-theoretic route to connecting knot contact geometry, HOMFLYPT recursions, and large-$N$ duality, with potential implications for understanding augmentation varieties and holonomic structures in knot theory.

Abstract

We prove a conjecture relating augmentation varieties to the large $N$ limit of Chern-Simons theory. Although this does not directly establish that the augmentation polynomial of a knot is the classical limit of a deformed $\hat{A}$-polynomial -- as suggested by Aganagić and Vafa -- it reduces the problem to characterizing certain algebraic properties of a module over the quantum torus, introduced in work of Gaiotto, Kannagi, and Sanjurjo. We term this the \emph{HOMFLYPT difference module}, which captures relations between the colored HOMFLYPT polynomials of different antisymmetric colorings. We demonstrate that the classical limit of this difference module for a knot is precisely the degree 0 abelianized knot contact homology of the knot, and we provide a natural extension of this result to links.

Knot contact homology as a planar limit of Chern-Simons theory

TL;DR

The paper develops a skein-theoretic, Deligne-style planar-limit framework to connect the augmentation varieties of knots and links with the large- limit of Chern-Simons theory. It introduces the HOMFLYPT difference module as a module over a quantum torus, built from heavy and light objects in a planar limit of the HOMFLYPT spider category, and shows that its degree-0 classical limit recovers abelianized knot contact homology (KCH). The central result is an isomorphism between KCH and the planar-limit invariant , with a conjectural identification of the augmentation ideal as the classical limit of the quantum-parameterized recursion ideal for colored HOMFLYPT polynomials, extending to links. Together these contributions provide an algebraic, skein-theoretic route to connecting knot contact geometry, HOMFLYPT recursions, and large- duality, with potential implications for understanding augmentation varieties and holonomic structures in knot theory.

Abstract

We prove a conjecture relating augmentation varieties to the large limit of Chern-Simons theory. Although this does not directly establish that the augmentation polynomial of a knot is the classical limit of a deformed -polynomial -- as suggested by Aganagić and Vafa -- it reduces the problem to characterizing certain algebraic properties of a module over the quantum torus, introduced in work of Gaiotto, Kannagi, and Sanjurjo. We term this the \emph{HOMFLYPT difference module}, which captures relations between the colored HOMFLYPT polynomials of different antisymmetric colorings. We demonstrate that the classical limit of this difference module for a knot is precisely the degree 0 abelianized knot contact homology of the knot, and we provide a natural extension of this result to links.
Paper Structure (13 sections, 12 theorems, 67 equations)

This paper contains 13 sections, 12 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Heavyskeinalgebras
  3. TheHOMFLYPTspider
  4. Heavyobjects
  5. Braidsandcups
  6. Planarlimit
  7. Tanglesandknots
  8. Label-shifting
  9. Comparisonwithknotcontacthomology
  10. Backgroundonknotcontacthomology
  11. Comparison
  12. Links
  13. Comparisonwiththeaugmentationideal

Key Result

Theorem A

Thebasechange$\mathsf{H}(L)$of$\mathsf{H}_q(L)$tothelocuswhere$q=1$(keeping$\nu_i,\Lambda_i$asformalvariables)isthedegree0partoftheabelianizedknotcontacthomology$\operatorname{KCH}(L)$undertheparameterchange: Inparticular,thebasechange$I(L)$oftheideal$I_q(L)$liesintheaugmentationideal$\mathsf{Au}_L\subset\mathbb{C}[U^{\pm1},\mu_i^{\pm1},\lambda_i^{\pm1}]$of$L$.

Theorems & Definitions (38)

  • Definition 1.1
  • Theorem A
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 28 more