Ribbonlength of torus knots
Brooke Brennan, Thomas W. Mattman, Roberto Raya, Dan Tating
Abstract
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).
Create a lesson
Related papers
Combinatorial Goussarov-Polyak-Viro Formulas for the Linking Number and Low Degree Coefficients of the Conway Polynomial
Nancy Scherich, Nathaniel Song
The flip symmetry on Khovanov-Rozansky homology
Hongjian Yang
Families of knots that cannot be made Legendrian parametrically
Javier Martínez-Aguinaga
Branched real projective structures on surfaces and geometrisation of representations
Gianluca Faraco, Nicholas Rungi
Generating the twist subgroup of the level 2 mapping class group of a non-orientable closed surface
Ryoma Kobayashi
H-cobordisms, infinite cyclic covers, and real Seiberg--Witten theory
Sungkyung Kang, JungHwan Park, Masaki Taniguchi