Relationships between braid length and the number of braid strands

Van Cott, Cornelia

doi:10.2140/agt.2007.7.181

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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

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 2000

Primary: 57M25

Secondary: 20F36

References

Publication

Received: 14 August 2006

Revised: 8 December 2006

Accepted: 11 January 2007

Published: 29 March 2007

Authors

Cornelia A Van Cott
Department of Mathematics Indiana University Bloomington IN 47405 USA
http://mypage.iu.edu/~cvancott/

Volume 7, issue 1 (2007)