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 [Formula: see text]-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.
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