Torsion in the Kauffman bracket skein module of a knot exterior
报告题目(Title)：Torsion in the Kauffman bracket skein module of a knot exterior
报告人(Speaker)：陈海苗（北京工商大学）
地点(Place)：后主楼1220
时间(Time)：2024年12月4日 10:00-11:00
邀请人(Inviter)：程志云
报告摘要
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.
