Recognizing destabilization, exchange moves and flypes

The Markov Theorem Without Stabilization (MTWS) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive there are three key isotopies that were identified and analyzed---destabilization, exchange moves and elementary braid preserving flypes. One of the critical open problems left in the wake of the MTWS is the "recognition problem"---determining when a given closed $n$-braid admits a specified move of the calculus. In this note we give an algorithmic solution to the recognition problem for these three key isotopies of the MTWS calculus. The algorithm is "directed" by a complexity measure that can be {\em monotonically simplified} by the application of "elementary moves".

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