Bi-Lipschitz characterization of space curves

TL;DR

This paper investigates how the bi-Lipschitz classification of space curves depends on local topological properties and shows that in higher dimensions, it is determined by singular points and generic projections into the plane.

Contribution

It extends the understanding of bi-Lipschitz classification from plane curves to space curves, highlighting the role of singular points and projections.

Findings

Bi-Lipschitz type of space curves is determined by singular points.

Generic projections into the plane preserve bi-Lipschitz classification.

Higher-dimensional space curves require different invariants than plane curves.

Abstract

In the paper \cite{renato} Renato Targino shows that bi-Lipschitz type of plane curve is determined by the local ambient topological properties of curves. Here we show that it is not longer true in higher dimensions. However we show that bi-Lipschitz type of space curves is determined by the number of singular points and by the local ambient topological type of a generic projection of such curves into the affine plane.