Noncrossing partitions for periodic braids

TL;DR

This paper characterizes elements in the super summit set of certain periodic braids using noncrossing partitions, providing a method to find conjugating elements and determine the set's size.

Contribution

It offers a combinatorial characterization of periodic braid elements in the dual Garside structure via noncrossing partitions, enabling explicit conjugation solutions.

Findings

Characterization of super summit set elements using noncrossing partitions

Explicit conjugating elements for periodic braids

Determination of super summit set size via zeta polynomial

Abstract

An element in Artin's braid group $B_n$ is called periodic if it has a power which lies in the center of $B_n$. The conjugacy problem for periodic braids can be reduced to the following: given a divisor $1\le d<n-1$ of $n-1$ and an element $\alpha$ in the super summit set of $\epsilon^d$, find $\gamma\in B_n$ such that $\gamma^{-1}\alpha\gamma=\epsilon^d$, where $\epsilon=(\sigma_{n-1}\cdots\sigma_1)\sigma_1$. In this article we characterize the elements in the super summit set of $\epsilon^d$ in the dual Garside structure by studying the combinatorics of noncrossing partitions arising from periodic braids. Our characterization directly provides a conjugating element $\gamma$. And it determines the size of the super summit set of $\epsilon^d$ by using the zeta polynomial of the noncrossing partition lattice.