A multiple conjugation quandle and handlebody-knots
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.
scientific article; zbMATH DE number 6522007