A Quantum invariant of Links in the Thickened Torus with Volume Conjecture Behavior

Joseph Boninger (New York)

Abstract

A growing body of evidence suggests that the asymptotic behavior of quantum invariants of links and 3-manifolds encodes geometric information. For example, the well-known Volume Conjecture of Kashaev, Murakami, and Murakami posits a relationship between the growth rate of the colored Jones polynomial of a link and the simplicial volume of its complement.

We define a quantum invariant of links in the thickened torus, which we call the toroidal colored Jones polynomial. Like the usual colored Jones polynomial, our invariant can be defined skein-theoretically or via the theory of operator invariants. Most interestingly, our toroidal colored Jones polynomial exhibits volume conjecture behavior. We prove volume convergence for the two-by-two square weave, and show that a suitably defined volume conjecture for our invariant implies the original Volume Conjecture.