BIMSA > BIMSA-Tsinghua Quantum Symmetry Seminar Torsion in the Kauffman bracket skein module of a knot exterior
Torsion in the Kauffman bracket skein module of a knot exterior
Organizers
Lin Zhe Huang , Zheng Wei Liu , Sébastien Palcoux , Yi Long Wang , Jin Song Wu
Speaker
Haimiao Chen
Time
Thursday, October 10, 2024 2:00 PM - 3:15 PM
Venue
A3-3-301
Online
Zoom 518 868 7656 (BIMSA)
Abstract
For a compact oriented $3$-manifold $M$, its {\it Kauffman bracket skein module} $\mathcal{S}(M)$ is defined as the quotient of the free $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-module generated by isotopy classes of framed links embedded in $M$ by the submodule generated by skein relations. It was known in 1990s that $\mathcal{S}(M)$ may admit torsion if $M$ contains an essential sphere or torus. A problem in ``Kirby's list" asks whether $\mathcal{S}(M)$ is free when $M$ does not contains an essential sphere or torus. We show that $\mathcal{S}(M)$ has infinitely many torsion elements when $M$ is the exterior of the $(a_1/b_1,a_2/b_2,a_3/b_4,a_4/b_4)$ Montesinos knot with each $b_i\ge 3$; in particular, $\mathcal{S}(M)$ is not free. Using surgery we can construct closed hyperbolic $3$-manifolds $N$ such that $\beta_1(N)=0$ and $\mathcal{S}(N)$ admits torsion.