Topology Seminar-Kauffman bracket skein modules and Chebyshev polynomials
Speaker: Rhea Palak Bakshi ( Institute for Theoretical Studies at ETH Zürich in Switzerland)
Place/time: , May 16, 2023; 4:00-5:00pm; room: Rome 771)
Title: Kauffman bracket skein modules and Chebyshev polynomials
Abstract: Skein modules were introduced by Józef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is known to be notoriously hard, especially over the ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years for KBSMs. We show that some of these conjectures, and even theorems, are not true. In this talk I will briefly discuss a counterexample to Marche’s generalisation of Witten’s conjecture. I will also give the exact structure of the KBSM off of the connected sum of two solid tori and show that it is isomorphic to the KBSM of a genus two handlebody modulo some specific handle sliding relations. Moreover, these handle sliding relations can be written in terms of Chebyshev polynomials. The last result is joint work with Thang Lê and Józef Przytycki.