Relationships between braid length and the number of braid strands
Abstract
For a knot K, let bn(K) be the minimum length of an n-stranded braid representative of K. Examples of knots exist for which bn(K) is a non-increasing function. We investigate the behavior of bn(K). We develop bounds on the function in terms of the genus of K, with stronger results for homogeneous knots and braid positive knots. For knots of nine or fewer crossings, we show that bn(K) is an increasing function and determine it completely.
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