A Quantum Invariant of Links in $T^2 \times I$ with Volume Conjecture Behavior

TL;DR

This paper introduces a new polynomial invariant for links in the thickened torus that exhibits volume conjecture behavior, with proofs for specific links and computational evidence for others, extending to virtual links.

Contribution

The paper defines the toroidal colored Jones polynomial $J_n^T$, proves the volume conjecture for the 2-by-2 square weave, and constructs it via two new methods, including for virtual links.

Findings

Proves volume conjecture for the square weave.

Provides computational evidence for other links.

First example of volume conjecture behavior in virtual links.

Abstract

We define a polynomial invariant $J_n^T$ of links in the thickened torus. We call $J^T_n$ the $n$th toroidal colored Jones polynomial, and show it satisfies many properties of the original colored Jones polynomial. Most significantly, $J_n^T$ exhibits volume conjecture behavior. We prove the volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions of $J_n^T$, one as a generalized operator invariant we call a pseudo-operator invariant, and another using the Kauffman bracket skein module of the torus. Finally, we show $J^T_n$ produces invariants of biperiodic and virtual links. To our knowledge, $J^T_n$ gives the first example of volume conjecture behavior in a virtual (non-classical) link.