Surface singularities in $R^4$: first steps towards Lipschitz knot theory

TL;DR

This paper explores the Lipschitz classification of surface singularities in four-dimensional space, revealing infinitely many Lipschitz classes for each knot type, thus advancing the understanding of Lipschitz knot theory.

Contribution

It establishes that for any knot in three-sphere, there are infinitely many Lipschitz equivalence classes of surface singularities in four-space sharing that knot as a link.

Findings

Infinitely many Lipschitz classes per knot type.

Lipschitz classification refines topological knot classification.

Surface singularities in R^4 can be distinguished by Lipschitz geometry.

Abstract

A link of an isolated singularity of a two-dimensional semialgebraic surface in $R^4$ is a knot (or a link) in $S^3$. Thus the ambient Lipschitz classification of surface singularities in $R^4$ can be interpreted as a bi-Lipschitz refinement of the topological classification of knots (or links) in $S^3$. We show that, given a knot $K$ in $S^3$, there are infinitely many distinct ambient Lipschitz equivalence classes of outer metric Lipschitz equivalent singularities in $R^4$ with the links topologically equivalent to $K$.