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

Boninger, Joe

doi:10.2140/agt.2023.23.1891

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A quantum invariant of links in $T^2 \times I$ with volume conjecture behavior

Joe Boninger

Algebraic & Geometric Topology 23 (2023) 1891–1934

DOI: 10.2140/agt.2023.23.1891

Abstract

We define a polynomial invariant $J_{n}^{T}$ of links in the thickened torus. We call $J_{n}^{T}$ 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 two-by-two 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 $J_{n}^{T}$ produces invariants of biperiodic and virtual links. To our knowledge, $J_{n}^{T}$ gives the first example of volume conjecture behavior in a virtual (nonclassical) link.

Keywords

Jones polynomial, volume conjecture, colored Jones polynomial, knots, virtual links, biperiodic links, torus, quantum invariants

Mathematical Subject Classification

Primary: 57K14, 81R50

References

Publication

Received: 12 April 2021

Revised: 1 October 2021

Accepted: 24 October 2021

Published: 14 June 2023

Authors

Joe Boninger
Department of Mathematics The Graduate Center, CUNY New York, NY United States

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Volume 23, issue 4 (2023)