For a knot K, let ℓ(K,n) be the minimum length of an n–stranded braid representative of K. Fixing a knot K, ℓ(K,n) can be viewed as a function of n, which we denote by ℓK(n). Examples of knots exist for which ℓK(n) is a nonincreasing function. We investigate the behavior of ℓK(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 ℓK(n) is eventually stable. We study the stable behavior of ℓK(n), with stronger results for homogeneous knots. For knots of nine or fewer crossings, we show that ℓK(n) is stable on all of its domain and determine the function completely.
Cornelia A Van Cott. Relationships between braid length and the number of braid strands. Algebraic & Geometric Topology 7 (2007) 181–196. DOI: 10.2140/agt.2007.7.181