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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

We define a polynomial invariant JnT of links in the thickened torus. We call JnT the n th toroidal colored Jones polynomial, and show it satisfies many properties of the original colored Jones polynomial. Most significantly, JnT 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 JnT, 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 JnT produces invariants of biperiodic and virtual links. To our knowledge, JnT gives the first example of volume conjecture behavior in a virtual (nonclassical) link.

Jones polynomial, volume conjecture, colored Jones polynomial, knots, virtual links, biperiodic links, torus, quantum invariants
Mathematical Subject Classification
Primary: 57K14, 81R50
Received: 12 April 2021
Revised: 1 October 2021
Accepted: 24 October 2021
Published: 14 June 2023
Joe Boninger
Department of Mathematics
The Graduate Center, CUNY
New York, NY
United States

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