Algebraic structures among virtual singular braids
type: Item
instance of: scholarly article
title: Algebraic structures among virtual singular braids
zbMATH Open document ID: 1548.20058
author: Carmen Caprau
author: Antonia Yeung
published in: La Matematica
publication date: 2024-09-27T00:00:00Z
review text: The classification of knots and links is equivalent to certain algebraic properties of classical braids as \textit{A. A. Markov} [Tr. Mat. Inst. Steklova 16, 3--53 (1945; Zbl 0061.02507)] showed. Similarly, the study of virtual knots, singular knots, virtual singular knots, and welded knots is intimately linked to that of virtual braids, singular braids, virtual singular braids, respectively.\N\NThe paper under review is devoted to the study of the algebraic properties of virtual singular braids. In particular, the authors show that the virtual singular braid monoid on \(n\) strands embeds in a group \(\mathrm{VSG}_{n}\), which they call the virtual singular braid group on \(n\) strands. The group \(\mathrm{VSG}_{n}\) contains a normal subgroup \(\mathrm{VSPG}_{n}\) of virtual singular pure braids and they show that \(\mathrm{VSG}_{n}\) is a semi-direct product of \(\mathrm{VSPG}_{n}\) and the symmetric group \(S_{n}\). The authors also represent \(\mathrm{VSPG}_{n}\) as a semi-direct product of \(n-1\) subgroups and they use these results to exhibit a normal form of words in the virtual singular braid group.
reviewed by: Egle Bettio
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57K12
zbMATH DE Number: 7919678
zbMATH Keywords: braid
zbMATH Keywords: Reidemeister-Schreier method
zbMATH Keywords: semi-direct product
zbMATH Keywords: virtual singular pure braid
MaRDI profile type: Publication
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DOI: 10.1007/S44007-024-00120-2
description: scientific article; zbMATH DE number 7919678