Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials

TL;DR

This paper develops a Legendrian Symplectic Field Theory framework for knot conormal tori, linking it to colored HOMFLY-PT polynomials and deriving recursion relations through algebraic elimination, advancing the understanding of knot invariants via symplectic and gauge theories.

Contribution

It introduces a novel SFT-based approach to relate knot contact homology with colored HOMFLY-PT polynomials and derives recursion relations using elimination theory.

Findings

HOMFLY-PT wave function is determined by SFT via induction.

Recursion relations for HOMFLY-PT polynomials are obtained through algebraic elimination.

Higher genus analogues of augmentation variety relations are established.

Abstract

We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large $N$ duality and Witten's connection between open Gromov-Witten invariants and Chern-Simons gauge theory, we relate the SFT of a link conormal to the colored HOMFLY-PT polynomials of the link. We present an argument that the HOMFLY-PT wave function is determined from SFT by induction on Euler characteristic, and also show how to, more directly, extract its recursion relation by elimination theory applied to finitely many noncommutative equations. The latter can be viewed as the higher genus counterpart of the relation between the augmentation variety and Gromov-Witten disk potentials established by Aganagic, Vafa, and the authors, and, from this perspective, our results can be seen as an SFT approach to quantizing the augmentation variety.