Algebraic & Geometric Topology 7 (2007) 181–196

DOI: 10.2140/agt.2007.7.181

arXiv: math.GT/0605476

Abstract

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 ℓ_K(n). Examples of knots exist for which ℓ_K(n) is a nonincreasing function. We investigate the behavior of ℓ_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 ℓ_K(n) is eventually stable. We study the stable behavior of ℓ_K(n), with stronger results for homogeneous knots. For knots of nine or fewer crossings, we show that ℓ_K(n) is stable on all of its domain and determine the function completely.

Keywords

knot theory, braid theory, braid index

Mathematical Subject Classification

Primary: 57M25

Secondary: 20F36

References

Publication

Received: 14 August 2006
Revised: 8 December 2006
Accepted: 11 January 2007
Published: 29 March 2007

Authors