Volume 7, issue 1 (2007)

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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
Abstract

For a knot $K$, let $\ell \left(K,n\right)$ be the minimum length of an $n$–stranded braid representative of $K$. Fixing a knot $K$, $\ell \left(K,n\right)$ can be viewed as a function of $n$, which we denote by ${\ell }_{K}\left(n\right)$. Examples of knots exist for which ${\ell }_{K}\left(n\right)$ is a nonincreasing function. We investigate the behavior of ${\ell }_{K}\left(n\right)$, 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 ${\ell }_{K}\left(n\right)$ is eventually stable. We study the stable behavior of ${\ell }_{K}\left(n\right)$, with stronger results for homogeneous knots. For knots of nine or fewer crossings, we show that ${\ell }_{K}\left(n\right)$ is stable on all of its domain and determine the function completely.

Keywords
knot theory, braid theory, braid index
Primary: 57M25
Secondary: 20F36
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/