Title: Relationships between braid length and the number of braid strands
Abstract: For a knot K , let `.K; n/ be the minimum length of an n-stranded braid representative of K .Fixing a knot K , `.K; n/ can be viewed as a function of n, which we denote by `K .n/.Examples of knots exist for which `K .n/ is a nonincreasing function.We investigate the behavior of `K .n/,developing bounds on the function in terms of the genus of K .The bounds lead to the conclusion that for any knot K the function `K .n/ is eventually stable.We study the stable behavior of `K .n/,with stronger results for homogeneous knots.For knots of nine or fewer crossings, we show that `K .n/ is stable on all of its domain and determine the function completely.