8字结不能解开

hbghlyj

如何证明8字结不等价于平凡结

吴振奎 - 数学中的美-上海教育出版社 (2002) p.83

hbghlyj

Purpose of the knot group
Let $K \subset S^3$ be the figure eight knot. One reason $K$ is not a trivial knot is because there is a surjective homomorphism from the group $\pi_1(S^3-K)$ onto the dihedral group of order 10. This is an exercises about the knot group that might be found in knot theory books, and I will leave it to you to look this up, or to figure it out using some presentation of $\pi_1(S^3-K)$ such as the Wirtinger presentation. This suffices to prove that the group $\pi_1(S^3-K)$ is not infinite cyclic, because the only groups onto which an infinite cyclic group can surject are other cyclic groups; but the dihedral group of order $10$ is not cyclic, it is not even abelian.

hbghlyj

Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was hyperbolic, by decomposing its complement into two ideal hyperbolic tetrahedra.
见The Geometry and Topology of Three-Manifolds

Let $M$ be the manifold obtained by removing the vertex.
It turns out that this manifold is homeomorphic with the complement of a figure-eight knot.

Youtube上有一个三集的视频详细讲了四维空间中的扭结。

Czhang271828

其实用基本群研究绳结的方法有时挺低效的, 研究 knot invariant 的工具还是挺多的. 一般绳结在幺半范畴的表示中用的比较多(具体应用领域如量子力学等), 这里的八字节就相当于 (strict) ribbon category 里的一个 twist, 确实无法通过 Reidemeister moves 消解.