Reducible braids and Garside theory

TL;DR

This paper demonstrates that reducible braids with minimal Garside-theoretical complexity are also geometrically simple, and provides a polynomial-time algorithm to determine their Nielsen-Thurston type based on cyclic sliding operations.

Contribution

It establishes a link between Garside-theoretical simplicity and geometric simplicity for reducible braids, and introduces an efficient algorithm for classifying braid types.

Findings

Reducible braids with minimal Garside complexity have obvious geometric reducibility.

A polynomial-time algorithm can find elements in the stabilized set of sliding circuits.

The algorithm's efficiency depends on a conjecture about cyclic sliding convergence.

Abstract

We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its conjugacy class which we call the stabilized set of sliding circuits, and if it is reducible, then its reducibility is geometrically obvious: it has a round or almost round reducing curve. Moreover, for any given braid, an element of its stabilized set of sliding circuits can be found using the well-known cyclic sliding operation. This leads to a polynomial time algorithm for deciding the Nielsen-Thurston type of any braid, modulo one well-known conjecture on the speed of convergence of the cyclic sliding operation.