The fundamental multiple conjugation quandle of a handlebody-link
Quandles were introduced in 1982 by \textit{D. Joyce} [J. Pure Appl. Algebra 23, 37--65 (1982; Zbl 0474.57003)] and \textit{S. V. Matveev} [Math. USSR, Sb. 47, 73--83 (1984; Zbl 0523.57006); translation from Mat. Sb., Nov. Ser. 119(161), No. 1, 78--88 (1982)] as an invariant for classical knots and links. Over the years, quandles have been investigated by various authors for constructing new invariants for knots and links. In the last years many papers about algebraic properties of quandles have appeared. The notion of a handlebody-link introduced in [\textit{A. Ishii}, Algebr. Geom. Topol. 8, No. 3, 1403--1418 (2008; Zbl 1151.57007)] generalizes the notion of a link. A multiple conjugation quandle introduced in [\textit{A. Ishii}, Topology Appl. 196, Part B, 492--500 (2015; Zbl 1348.57024)] is an algebraic system whose axioms are derived from the Reidemeister moves for handlebody-link diagrams. Hence, the notion of a multiple conjugation quandle generalizes the notion of a quandle. By analogy with how for each link one can construct a quandle, which is an invariant for links, the author of the paper under review developed a procedure of how for each handlebody-link to construct a multiple conjugation quandle, which is also an invariant. The main problem of constructing such an invariant was that the quotient of multiple conjugation quandles does not form a multiple conjugation quandle itself. The author solves this problem by introducing the notion of a quasi multiple conjugation quandle and its multiple conjugation quandle closure. Using the multiple conjugation quandle closure, the author succeeds in defining a presentation of a multiple conjugation quandle of a handlebody-link, which is an invariant for handlebody-links.
fundamental multiple conjugation quandle
scientific article; zbMATH DE number 7470018