Document Zbl 1429.57006 - zbMATH Open

A handlebody-knot is a handlebody embedded in \(S^3\). Similar as knots, two \(S^1\)-oriented (a fixed orientation for the core \(S^1\) of each genus one component) handlebody-knots with the same genus can be connected by finitely many crossing changes. The paper under review is devoted to studying the Gordian distance of two \(S^1\)-oriented handlebody-knots, i.e. the minimal number of crossing changes needed to convert one handlebody-knot into the other one.
The main tool of this paper is the Alexander biquandle coloring invariant. Recall that a biquandle is a set with two binary operations which satisfy some axioms motivated by Reidemeister moves. It was first noticed by A. Ishii et al. [Ill. J. Math. 57, No. 3, 817–838 (2013; Zbl 1306.57011)] that one can combine the biquandle idea with the fundamental group of the handlebody-knot complement. It turns out that the notion of \(G\)-family of quandles plays an important role in the study of handlebody-knot theory. Here \(G\) denotes a group. Later, this idea was extended to \(G\)-families of biquandles. By using the \(\mathbb{Z}_m\)-family of Alexander biquandles, the author proves that one can use the difference of the rank of colorings of two handlebody-knot diagrams to obtain a lower bound for the Gordian distance. As an important application, the author shows that each positive integer \(n\) can be realized as the Gordian distance of two handlebody-knots.

MSC:

57K10	Knot theory
57K12	Generalized knots (virtual knots, welded knots, quandles, etc.)
57M15	Relations of low-dimensional topology with graph theory

References:

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