Ribbon crossing numbers, crossing numbers, and Alexander polynomials
type: Item
instance of: scholarly article
title: Ribbon crossing numbers, crossing numbers, and Alexander polynomials
zbMATH Open document ID: 1404.57043
author: Tomoyuki Yasuda
published in: Topology and its Applications
publication date: 2018-08-24T00:00:00Z
review text: Ribbon \(2\)-knotted objects are locally flat embeddings of surfaces in \(4\)-space which bound immersed \(3\)-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. Let \(K^2\) be a ribbon \(2\)-knot. The ribbon crossing number, denoted here by \(r-cr(K^2)\) is a numerical invariant of the ribbon \(2\)-knot \(K^2\). It is known that the degree (the difference of between the maximal degree and the minimal degree) of the Alexander polynomial of \(K^2\) is less than or equal to \(r-cr(K^2)\). In the paper under review the author shows that \(r-cr(K^2)\) is estimated by coefficients of the Alexander polynomial of \(K^2\). As an application, the author obtains an estimation of the crossing number, denoted by \(cr(k^1)\), for a classical knot \(k^1\).
reviewed by: Michał Jabłonowski
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6924615
zbMATH Keywords: ribbon 2-knot
zbMATH Keywords: knot
zbMATH Keywords: ribbon crossing number
zbMATH Keywords: crossing number
zbMATH Keywords: Alexander polynomial
MaRDI profile type: Publication
full work available at URL: j.topol.2018.07.007
OpenAlex ID: W2867887118
cites work: Q4136199
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cites work: Some problems in knot theory
Wikidata QID: Q129555564
DOI: 10.1016/J.TOPOL.2018.07.007
description: scientific article; zbMATH DE number 6924615