Recognizing trace graphs of closed braids

TL;DR

This paper introduces a method to associate trace graphs to closed braids in a solid torus, enabling recognition of braid isotopy through graph moves with polynomial-time algorithms.

Contribution

It presents a novel approach linking closed braids to trace graphs and provides an efficient polynomial-time recognition algorithm for fixed strand numbers.

Findings

Trace graphs uniquely represent closed braids up to isotopy and trihedral moves.

Recognition of trace graphs is polynomial-time for fixed number of strands.

The method facilitates braid classification using graph transformations.

Abstract

To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetraherdal moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral moves in polynomial time with respect to the braid length.