2011/01/18 by A. Stoimenow, Stoimenow, A. · 3 citations
Mathematics · #53D10 #57M15 (secondary) #57M25 (primary) #57N10 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:53D10 #msc:57M15 #msc:57M25 #msc:57N10
paper · pdf · doi:10.48550/arxiv.1101.3390
78pages, 14 figures
arxiv created 2011/01/18 · openalex publication_date 2011/01/18 · arxiv updated 2015/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We continue the study of the genus of knot diagrams, deriving a new description of generators using Hirasawa's algorithm. This description leads to good estimates on the maximal number of crossings of generators and allows us to complete their classification for knots of genus 4. As applications of the genus 4 classification, we establish non-triviality of the skein polynomial on k-almost positive knots for k≤ 4, and of the Jones polynomial for k≤ 3. For k≤ 4, we classify the occurring achiral knots, and prove a trivializability result for k-almost positive unknot diagrams. This yields also estimates on the number of unknotting Reidemeister moves. We describe the positive knots of signature (up to) 4. Using a study of the skein polynomial, we prove the exactness of the Morton-Williams-Franks braid index inequality and the existence of a minimal string Bennequin surface for alternating knots up to genus 4. We also prove for such knots conjectures of Hoste and Fox about the roots and coefficients of the Alexander polynomial.