On a commuting graph of tiny braids

Abstract

A positive braid is called tiny when it is a product of generators and each generator being used at most once. We will present a commuting criterion for tiny braids and explore the structure of the centralizer of a tiny braid in the set of positive braid monoids. We will also discuss various combinatorial properties of the centralizer and a few invariants of an associated commuting graph. Finally, we will provide affirmative answers to a couple of open questions related to the commuting graph of tiny braids.