2004/10/26 by Brooke Brennan, Thomas W. Mattman, Brennan, Brooke +5 · 3 citations
Mathematics · #Geometric and Algebraic Topology #math.GT #msc:57M25
paper · pdf · doi:10.48550/arxiv.math/0410565
11 pages, 8 figures
arxiv created 2004/10/26 · arxiv updated 2009/12/01
Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realised by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby bounding the Ribbonlength of the knots. In particular, we give evidence that the closed (respectively, truncation) Ribbonlength of a (q+1,q) torus knot is (2q+1)cot(π/(2q+1)) (resp., 2q cot(π/(2q+1))). Using these calculations, we provide the bounds c1 ≤ 2/πand c2 ≥ 5/3 cot(π/5) for the constants c1 and c2 that relate Ribbonlength R(K) and crossing number C(K) in a conjecture of Kusner: c1 C(K) ≤ R(K) ≤ c2 C(K).