2015/11/30 by Kathryn A. Bryant, Kathryn Bryant · 8 citations
Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #Knot invariant #Knot theory #Ribbon #Tricolorability #Twist #math.GT
paper · pdf · doi:10.2140/agt.2017.17.3621
published in Algebraic & Geometric Topology 17(6), 3621-3664 (Mathematical Sciences Publishers) · 31 pages, 9 figures. Version 2 includes results for pretzel knots with single-twists (strands with twisting parameter 1 or -1), which were not addressed in version 1. Version 2 also includes one new figure and fixes a few typos
arxiv created 2016/01/18 · openalex created_date 2016/06/24 · openalex publication_date 2017/10/04 · arxiv updated 2018/03/16 · openalex updated_date 2026/08/08
A pretzel knot K is called odd if all its twist parameters are odd, and mutant ribbon if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: All slice, odd, 5-stranded pretzel knots are mutant ribbon. We do this in stages by first showing that 5-stranded pretzel knots having twist parameters with all the same sign or with exactly one parameter of a different sign have infinite order in the topological knot concordance group, and thus in the smooth knot concordance group as well. Next, we show that any odd, 5-stranded pretzel knot with zero pairs or with exactly one pair of canceling twist parameters is not slice.