A multiple conjugation quandle and handlebody-knots
type: Item
instance of: scholarly article
title: A multiple conjugation quandle and handlebody-knots
zbMATH Open document ID: 1348.57024
author: Atsushi Ishii
published in: Topology and its Applications
publication date: 2015-12-18T00:00:00Z
review text: The author investigates a notion of multiple conjugation quandle and gives some examples. This algebraic structure is motivated and derived from the combinatorial structure of handlebody-knots. The author previously studied Reidemeister moves for handlebody-knots and a coloring invariant in [Algebr. Geom. Topol. 8, No. 3, 1403--1418 (2008; Zbl 1151.57007)] which was generalized in [\textit{A. Ishii} and \textit{M. Iwakiri}, Can. J. Math. 64, No. 1, 102--122 (2012; Zbl 1245.57014)] and [\textit{A. Ishii} et al., Ill. J. Math. 57, No. 3, 817--838 (2013; Zbl 1306.57011)]. The multiple conjugation quandle structure studied in the paper under review is a structure that mixes a group and a quandle: the operation of the quandle structure is used to describe the coloring condition at a crossing, while the multiplication of the group structure is used to describe the coloring condition at a vertex (see the precise definitions and the diagrammatics in the paper). Quandles with good involutions were defined by \textit{S. Kamada} [in: Intelligence of low dimensional topology 2006, Hiroshima, Japan July 22--26, 2006. Hackensack, NJ: World Scientific. 101--108 (2007; Zbl 1145.57008)] in order to study colorings for unoriented knots and surface-knots. The author of the paper under review proves that if one starts from a quandle with good involution and adds some conditions in order to be able to use the newer structure for colorings of handlebody-links then the obtained structure is the multiple conjugation quandle.
reviewed by: Mohamed Elhamdadi
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 6522007
zbMATH Keywords: quandle
zbMATH Keywords: handlebody-knot
zbMATH Keywords: algebra
MaRDI profile type: Publication
full work available at URL: j.topol.2015.05.029
OpenAlex ID: W916920730
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DOI: 10.1016/J.TOPOL.2015.05.029
description: scientific article; zbMATH DE number 6522007