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

TL;DR

This paper establishes a connection between augmentation varieties and the large N limit of Chern-Simons theory, linking knot contact homology to quantum algebraic structures and colored HOMFLYPT polynomials.

Contribution

It introduces the HOMFLYPT difference module and proves its classical limit equals the abelianized knot contact homology, advancing understanding of knot invariants and their algebraic properties.

Findings

Augmentation varieties relate to the large N limit of Chern-Simons theory.

The HOMFLYPT difference module captures relations between colored HOMFLYPT polynomials.

Classical limit of the difference module equals the abelianized knot contact homology.

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 Aganagi\'c 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.