A quantum invariant of links in T2× I with volume conjecture behavior

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 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 $T^2 \times I$, defining what we call a pseudo-operator invariant.