The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings

TL;DR

This paper explores the deep connection between perturbative knot invariants and topological string free energies on character varieties, extending the correspondence beyond leading order using recursion relations and specific models for hyperbolic and torus knots.

Contribution

It advances the understanding of the knot-topological string correspondence by analyzing higher-order terms and applying topological recursion to specific knot complements.

Findings

Confirmed the correspondence up to fourth order for certain knots.

Found trivial recursion relations for torus knots.

Extended the volume conjecture and AJ conjecture beyond leading order.

Abstract

We study the relation between perturbative knot invariants and the free energies defined by topological string theory on the character variety of the knot. Such a correspondence between SL(2;C) Chern-Simons gauge theory and the topological open string theory was proposed earlier on the basis of the volume conjecture and AJ conjecture. In this paper we discuss this correspondence beyond the subleading order in the perturbative expansion on both sides. In the computation of the perturbative invariants for the hyperbolic 3-manifold, we adopt the state integral model for the hyperbolic knots, and the factorized AJ conjecture for the torus knots. On the other hand, we iteratively compute the free energies on the character variety using the Eynard-Orantin topological recursion relation. We check the correspondence for the figure eight knot complement and the once punctured torus bundle over S^1 with the holonomy L^2R up to the fourth order. For the torus knots, we find trivial the recursion relations on both sides.