论文标题
Lipschitz表面细菌的几何形状，$ \ Mathbb {r}^4 $：公制结
Lipschitz geometry of surface germs in $\mathbb{R}^4$: metric knots
论文作者
论文摘要
在$ \ mathbb {r}^4 $中的二维半格式表面的孤立奇点的链接是$ s^3 $中的拓扑结（或链接）。我们研究了$ \ mathbb {r}^4 $的环境Lipschitz几何形状与结理论之间的连接。也就是说，对于任何打结$ k $，我们在$ \ mathbb {r}^4 $中构建一个表面$ x_k $，这样：$ x_ {k {k} $的链接是一个琐碎的结；细菌$ x_k $是所有$ k $的外部bi-lipschitz;两个细菌$ x_ {k} $和$ x_ {k'} $是环境bi-lipschitz等效的，只有当结$ k $和$ k'$是同位素时。我们表明，琼斯多项式可用于识别$ \ mathbb {r}^4 $中的环境BI-LIPSCHITZ，即使它们在拓扑上是微不足道和外部bi-lipschitz等效的，也可以识别它们。
A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in $\mathbb{R}^4$ is a topological knot (or link) in $S^3$. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in $\mathbb{R}^4$ and the knot theory. Namely, for any knot $K$, we construct a surface $X_K$ in $\mathbb{R}^4$ such that: the link at the origin of $X_{K}$ is a trivial knot; the germs $X_K$ are outer bi-Lipschitz equivalent for all $K$; two germs $X_{K}$ and $X_{K'}$ are ambient bi-Lipschitz equivalent only if the knots $K$ and $K'$ are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in $\mathbb{R}^4$, even when they are topologically trivial and outer bi-Lipschitz equivalent.
