Relationships between braid length and the number of braid strands

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.