Exchangeability and Non-Conjugacy of Braid Representatives

Abstract

We obtain some fairly general conditions on the linking numbers and geometric properties of a link, under which it has infinitely many conjugacy classes of n-braid representatives if and only if it has one admitting an exchange move. We investigate a symmetry pattern of indices of conjugate iterated exchanged braids. We then develop a test based on the Burau matrix showing examples of knots admitting no minimal exchangeable braids, admitting non-minimal non-exchangeable braids, and admitting both minimal exchangeable and minimal non-exchangeable braids. This in particular proves that conjugacy, exchange moves and destabilization do not suffice to simplify braid representatives of a general link. © 2021 World Scientific Publishing Company.