Twin groups of virtual 2-bridge knots and almost classical knots
type: Item
instance of: scholarly article
zbMATH Open document ID: 1256.57009
DOI: 10.1142/S0218216512500952
author: Takuji Nakamura
author: Yasutaka Nakanishi
author: Shin Satoh
author: Yumi Tomiyama
publication date: 2012-08-30T00:00:00Z
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57Q35
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 6074275
zbMATH Keywords: virtual knot
zbMATH Keywords: knot group
zbMATH Keywords: Wirtinger presentation
zbMATH Keywords: Gauss diagram
zbMATH Keywords: welded knot
zbMATH Keywords: Miyazawa polynomial
MaRDI profile type: Publication
cites work: VIRTUAL BRIDGE NUMBER ONE KNOTS
cites work: VIRTUAL CROSSING NUMBER AND THE ARROW POLYNOMIAL
cites work: The braid-permutation group
cites work: Finite-type invariants of classical and virtual knots
cites work: A SEQUENCE OF DEGREE ONE VASSILIEV INVARIANTS FOR VIRTUAL KNOTS
cites work: BRIDGE PRESENTATIONS OF VIRTUAL KNOTS
cites work: THE POLE DIAGRAM AND THE MIYAZAWA POLYNOMIAL
cites work: ABSTRACT LINK DIAGRAMS AND VIRTUAL KNOTS
cites work: FORBIDDEN MOVES UNKNOT A VIRTUAL KNOT
cites work: Virtual knot theory
cites work: A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR VIRTUAL KNOTS AND LINKS
cites work: UNKNOTTING VIRTUAL KNOTS WITH GAUSS DIAGRAM FORBIDDEN MOVES
cites work: On the crossing numbers of a virtual knot
cites work: CROWELL'S DERIVED GROUP AND TWISTED POLYNOMIALS
full work available at URL: s0218216512500952
OpenAlex ID: W2001178714
title: Twin groups of virtual 2-bridge knots and almost classical knots
published in: Journal of Knot Theory and its Ramifications
review text: The Wirtinger presentation of a knot group is a combinatorial recipe for computing the fundamental group of the complement of a knot from a knot diagram using a base point above the plane of the knot diagram. For classical knots, the groups obtained using base points above and below the plane of the knot diagram are isomorphic, but for virtual knots these groups can be different. In this paper the authors consider the question of which pairs of groups of a certain form can be realized as the upper and lower knot groups of a virtual knot, obtaining a sufficient condition. A virtual knot is \textit{almost classical} if it has a diagram locally admitting an Alexander numbering; the authors give a necessary condition for a virtual knot to be almost classical in terms of the Jones and Miyazawa polynomials and as an application, determine which of the virtual knots in the virtual knot table on the knot atlas are almost classical.
reviewed by: Sam Nelson
description: scientific article; zbMATH DE number 6074275