The Turaev genus of an adequate knot
type: Item
instance of: scholarly article
title: The Turaev genus of an adequate knot
zbMATH Open document ID: 1188.57006
author: Tetsuya Abe
published in: Topology and its Applications
publication date: 2009-11-05T00:00:00Z
review text: The author shows in this paper that if a knot \(K\) admits an adequate diagram \(D\), then the Turaev genus of the knot \(K\) is the Turaev genus of the diagram \(D\), and they are equal to \(w_{Kh}(K)-2\), where \(w_{Kh}(K)\) is the width of the Khovanov homology of \(K\). Furthermore, if \(K'\) is a knot which has a mutant of \(D\), then the Turaev genus of \(K'\) is the same as that of \(K\). The connected sum of two adequate knots is adequate, and the author shows that the Turaev genus of the connected sum of two adequate knots is the sum of the Turaev genera of these two knots. The author asks whether the last two results could be true for all knots. Finally it is shown that an \(n\)-semi-alternating knot has Turaev genus \(n\), this gives the first examples of adequate knots of genus greater than 1.
reviewed by: Michael C. Tsau
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 5625470
zbMATH Keywords: adequate knot
zbMATH Keywords: Khovanov homology
zbMATH Keywords: Turaev genus
zbMATH Keywords: Turaev surface
MaRDI profile type: Publication
full work available at URL: j.topol.2009.07.020
OpenAlex ID: W2029816803
cites work: AN ESTIMATION OF THE ALTERNATION NUMBER OF A TORUS KNOT
cites work: THE DEALTERNATING NUMBER AND THE ALTERNATION NUMBER OF A CLOSED 3-BRAID
cites work: Almost alternating links
cites work: Q4820380
cites work: THE SPREAD AND EXTREME TERMS OF JONES POLYNOMIALS
cites work: ARC INDEX OF NON-ALTERNATING LINKS
cites work: Graphs on surfaces and Khovanov homology
cites work: Q4829661
cites work: The Jones polynomial and graphs on surfaces
cites work: State models and the Jones polynomial
cites work: A categorification of the Jones polynomial
cites work: Patterns in Knot Cohomology, I
cites work: An endomorphism of the Khovanov invariant
cites work: Q4382059
cites work: Some links with non-trivial polynomials and their crossing-numbers
cites work: On knot Floer width and Turaev genus
cites work: Additional Gradings in Khovanov Homology
cites work: Jones polynomials and classical conjectures in knot theory
cites work: Union and tangle
cites work: Knot Floer homology, genus bounds, and mutation
cites work: Hard to identify (non-)mutations
cites work: Khovanov homology of torus links
cites work: A spanning tree expansion of the Jones polynomial
cites work: On the Kauffman polynomial of an adequate link
cites work: A simple proof of the Murasugi and Kauffman theorems on alternating links
cites work: A spectral sequence for Khovanov homology with an application to (3,q)-torus links
DOI: 10.1016/J.TOPOL.2009.07.020
description: scientific article; zbMATH DE number 5625470