# Isotopy and homeomorphism of closed surface braids

## TL;DR

This paper classifies closed surface braids in orientable surfaces up to isotopy and homeomorphism, revealing algebraic conditions involving surface braid groups and the mapping class group.

## Contribution

It provides a comprehensive classification of closed surface braids using algebraic structures, clarifying the relationship between isotopy, homeomorphism, and group actions.

## Key findings

- Braids close to isotopic links are conjugate in the braid group.
- Braids close to homeomorphic links are in the same orbit under the mapping class group action.
- Classification applies to surfaces of positive genus with specific indeterminacy for sphere braids.

## Abstract

The closure of a braid in a closed orientable surface $\Sigma$ is a link in $\Sigma\times S^1$. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its center.