On reduction curves and Garside properties of braids

TL;DR

This paper investigates the structure of braids through reduction curves, explores how cyclic sliding affects their normal forms, and provides examples of braids with exponentially large sliding circuit sets.

Contribution

It introduces a new approach to decomposing braids using reduction curves and analyzes the impact of cyclic sliding on their normal forms, with implications for braid complexity.

Findings

Decomposition of braids via reduction curves differs from Thurston's method.

Cyclic sliding influences the normal form of braids and their components.

Constructed braids with exponential size of sliding circuits and ultra summit sets.

Abstract

In this paper we study the reduction curves of a braid, and how they can be used to decompose the braid into simpler ones in a precise way, which does not correspond exactly to the decomposition given by Thurston theory. Then we study how a cyclic sliding (which is a particular kind of conjugation) affects the normal form of a braid with respect to the normal forms of its components. Finally, using the above methods, we provide the example of a family of braids whose sets of sliding circuits (hence ultra summit sets) have exponential size with respect to the number of strands and also with respect to the canonical length.