Monotonic Simplification and Recognizing Exchange Reducibility

In [BM4] 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 three isotopies of the MTWS calculus—destabilization, exchange moves and braid preserving flypes. The algorithm is “directed” by a complexity measure that can be monotonic simplified by that application of elementary moves.