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

Greater Washington Topology Seminar

The following is a list of confirmed speakers and dates for the Greater (George) Washington Topology Seminar seminar: 

1. March 5, 2021: Maciej Mroczkowski (University of Gdańsk)

2. March 12, 2021: Victoria Lebed (Université de Caen - Normandie )

3. March 26, 2021: Dirk Schütz (University of Durham)

4. April 2, 2021: Peter Samuelson (University of California, Riverside)

5. April 9, 2021: Krzysztof Putyra (Universität Zürich)

6. April 16, 2021: Michal Jablonowski (University of Gdańsk)

7. April 23, 2021: Nathan Geer (Utah State University)

8. April 30, 2021: Joseph Boninger (CUNY)

9. May 7, 2021: William Goldman (University of Maryland, College Park) 

10. May 14, 2021: Wade Bloomquist (Georgia Tech)


The next talk will be by Joseph Boninger and will take place on April 30, 2021 from 1:00pm to 2:00pm EDT. 

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

Abstract: The well-known Volume Conjecture of Kashaev, Murakami, and Murakami posits that the growth rate of the colored Jones polynomial of a link encodes the simplicial volume of its complement. Analogous conjectures have been made for other quantum invariants of links and three-manifolds, and a growing body of evidence suggests a relationship between these invariants and three-dimensional geometry.

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, via the skein module of the torus, or using 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.

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Topic: GW Topology Seminar 

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