Noncrossing partitions for periodic braids

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.