The Infimum, Supremum, and Geodesic Length of a Braid Conjugacy Class

Abstract

Algorithmic solutions to the conjugacy problem in the braid groups Bn,n =2, 3, 4, … were given in earlier work. This note concerns the computation of two integer class invariants, known as “inf” and “sup.” A key issue in both algorithms is the number m of times one must “cycle” (resp. “decycle”) in order to either increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the class. Our main result is to prove that m is bounded above by ((n2−n)/2)−1 in the situation stated by E. A. Elrifai and H. R. Morton (1994, Quart. J. Math. Oxford45, 479–497) and by n−2 in the situation stated by authors (1998, Adv. Math.139, 322–353). It follows immediately that the computation of inf and sup is polynomial in both word length and braid index, in both algorithms. The integers inf and sup determine (but are not determined by) the shortest geodesic length for elements in a conjugacy class, and so we also obtain a polynomial-time algorithm for computing this length.