Geodesic
Relationships between braid length and the number of braid strands
Van Cott, Cornelia1
1Department of Mathematics, Indiana University, Bloomington IN 47405, USA
Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 181-196
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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 ℓK(n). Examples of knots exist for which ℓK(n) is a nonincreasing function. We investigate the behavior of ℓK(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 ℓK(n) is eventually stable. We study the stable behavior of ℓK(n), with stronger results for homogeneous knots. For knots of nine or fewer crossings, we show that ℓK(n) is stable on all of its domain and determine the function completely.

DOI : 10.2140/agt.2007.7.181
Keywords: knot theory, braid theory, braid index

Van Cott, Cornelia  1

1 Department of Mathematics, Indiana University, Bloomington IN 47405, USA 
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Van Cott, Cornelia. Relationships between braid length and the number of braid strands. Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 181-196. doi: 10.2140/agt.2007.7.181

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