Torsion in the Kauffman bracket skein module of a knot exterior
演讲者
陈海苗
时间
2024年10月10日 14:00 至 15:15
地点
A3-3-301
线上
Zoom 518 868 7656
(BIMSA)
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.