Lipschitz geometry of surface germs in $\mathbb{R}^4$: metric knots

TL;DR

This paper explores the Lipschitz geometry of surface germs in four-dimensional space, establishing a link with knot theory and demonstrating how knot invariants like the Jones polynomial can distinguish non-equivalent surface germs.

Contribution

It constructs surface germs in $\,\mathbb{R}^4$ associated with knots, showing their Lipschitz equivalence classes correspond to knot isotopy classes, and uses knot invariants to distinguish them.

Findings

Surface germs are outer bi-Lipschitz equivalent regardless of the knot.

Germs are ambient bi-Lipschitz equivalent only if the knots are isotopic.

Jones polynomial can distinguish non-equivalent surface germs.

Abstract

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.