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Relationships between braid length and the number of braid strands

Cornelia A Van Cott

Algebraic & Geometric Topology 7 (2007) 181–196

arXiv: math.GT/0605476


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.

knot theory, braid theory, braid index
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 20F36
Received: 14 August 2006
Revised: 8 December 2006
Accepted: 11 January 2007
Published: 29 March 2007
Cornelia A Van Cott
Department of Mathematics
Indiana University
Bloomington IN 47405