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

Abstract

We define a polynomial invariant J T n of links in the thickened torus. We call J T n the nth toroidal colored Jones polynomial, and show it satisfies many properties of the original colored Jones polynomial. Most significantly, J T n 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 T n , one using operator invariants and another using the Kauffman bracket skein module of the torus. In the process we generalize the theory of operator invariants to links in T 2 × I, defining what we call a pseudo-operator invariant.