Noncrossing partitions for periodic braids

Abstract

An element in Artin's braid group Bn is called periodic if it has a power that lies in the center of Bn. The conjugacy problem for periodic braids can be reduced to the following: given a divisor 1⩽d<n−1 of n−1 and an element α in the super summit set of εd, find γ∈Bn such that γ−1αγ=εd, where ε=(σn−1⋯σ1)σ1.In this article we characterize the elements in the super summit set of εd in the dual Garside structure by studying the combinatorics of noncrossing partitions arising from periodic braids. Our characterization directly provides a conjugating element γ. And it determines the size of the super summit set of εd by using the zeta polynomial of the noncrossing partition lattice.