Noncrossing partitions for periodic braids

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 ${ϵ}^d$, find $\gamma \in B_n$ such that ${\gamma }^{-1}\alpha \gamma ={ϵ}^d$, where $ϵ=({\sigma }_{n-1}\cdots {\sigma }_1){\sigma }_1$. In this article we characterize the elements in the super summit set of ${ϵ}^d$ 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 ${ϵ}^d$ by using the zeta polynomial of the noncrossing partition lattice.