How to read the length of a braid from its curve diagram

TL;DR

This paper establishes a precise relationship between the Garside length of a braid and a winding-number invariant derived from its curve diagram, providing a new geometric perspective on braid length.

Contribution

It introduces a novel invariant based on the curve diagram that exactly matches the Garside length of a braid, linking algebraic and geometric braid properties.

Findings

Garside length equals a winding-number invariant

Provides a geometric method to determine braid length

Bridges algebraic and topological braid invariants

Abstract

We prove that the Garside length a braid is equal to a winding-number type invariant of the curve diagram of the braid.