Isotopy equivalence of analytic branches in $({\mathbb C}^n,0)$

TL;DR

This paper proves that analytic branches in complex n-space with identical dual resolution graphs are ambient isotopic outside the origin, advancing understanding of their topological classification.

Contribution

It establishes a sufficient condition for ambient isotopy of analytic branches based on their dual resolution graphs, with a partial converse.

Findings

Analytic branches with the same dual resolution graph are ambient isotopic outside the origin.

A weaker form of the converse statement is also demonstrated.

The results contribute to the topological classification of complex analytic branches.

Abstract

We prove that two analytic branches in $(\mathbb{C}^n,0)$ whose dual resolution graph is the same admit an ambient isotopy which is smooth outside the origin. A weaker version of the converse is also proved.