Biquandles and virtual links
type: Item
instance of: scholarly article
title: Biquandles and virtual links
zbMATH Open document ID: 1063.57006
author: Roger Fenn
author: Louis Kauffman
author: Mercedes Jordán-Santana
published in: Topology and its Applications
publication date: 2005-02-22T00:00:00Z
review text: The axioms of quandle theory may be understood as arising from knot theory by taking arcs as generators and requiring the resulting algebraic structure to be unchanged by Reidemeister moves. If we replace ``arcs'' with ``semi-arcs'' obtained by dividing knot diagrams at both over- and under-crossings, the resulting algebraic structure is known as a {biquandle}. In this paper, biquandles are obtained from {biracks,} which are algebraic structures derived from {switches}, algebraic solutions to the Yang-Baxter equation. Every birack defines a representation of the virtual braid group \(VB_n\), with resulting invariants of virtual knots. Some examples of switches and their resulting biracks are given, along with their applications to virtual knot theory. In particular, the fundamental biquandle is conjectured to be a complete invariant of virtual knots up to mirror image.
reviewed by: Sam Nelson
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 2136837
zbMATH Keywords: racks
zbMATH Keywords: quandles
zbMATH Keywords: virtual knots
MaRDI profile type: Publication
full work available at URL: j.topol.2004.06.008
OpenAlex ID: W2015563301
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DOI: 10.1016/J.TOPOL.2004.06.008
description: scientific article; zbMATH DE number 2136837