arXiv subjects

Katsunori Arai

Publications and source records attributed to Katsunori Arai.

5 recordsLinked to original sources

arXiv2026

A condition of admissibility for generalized Alexander quandles

A typical example of a quandle is the conjugation quandle. A quandle is said to be admissible if it is isomorphic to a conjugation quandle. We study the admissibility problem for quandles, that is, determining whether a given quandle is admissible. In particular, we focus on generalized Alexander quandles, which are groups equipped with quandle structures defined by group automorphisms. In this paper, we provide a sufficient condition for admissibility in terms of geometric properties of quandles: namely, if every antipodal set in each algebraically connected component consists of a single point, then the quandle is admissible. This result generalizes previous results on generalized Alexander quandles.

math.GT↗

arXiv2026

Detecting non-admissibility of quandles via colorings

A quandle is an algebraic system whose axioms are motivated by Reidemeister moves in knot theory. A typical example is a conjugation quandle arising from a group. A quandle is said to be admissible if it is isomorphic to a conjugation quandle. Admissible quandles often yield knot invariants that coincide with those derived from the knot group, whereas nonadmissible quandles may produce genuinely new invariants. In this sense, it is important to construct non-admissible quandles. In this paper, we provide criteria for determining whether given quandles are admissible by colorings of (1, 1)-tangles. As an application, we construct numerous examples of non-admissible quandles by analyzing simple tangles obtained from the Hopf link and the trefoil knot.

math.GT↗

arXiv2025

Infinitely many pairs of spatial surfaces

A multiple group rack (MGR) is an algebraic system which is used to construct invariants of spatial surfaces, which are compact surfaces embedded in the $3$-sphere $S^{3}$. Seifert surfaces for links are spatial surfaces. In this paper, we present an infinitely many pairs of Seifert surfaces for each link, where each pair satisfies the following condtions: (i) their regular neighborhoods in $S^{3}$ are ambiently isotopic, (ii) their Seifert matrices are unimodularly congruent, and (iii) the two Seifert surfaces are not ambiently isotopic. In order to prove (iii), we distinguish the Seifert surfaces using the above invariants.

math.GT↗

arXiv2025

A construction of multiple group racks

A multiple group rack is a rack which is a disjoint union of groups equipped with a binary operation satisfying some conditions. It is used to define invariants of spatial surfaces, i.e., oriented compact surfaces with boundaries embedded in the $3$-sphere $S^{3}$. A $G$-family of racks is a set with a family of binary operations indexed by the elements of a group $G$. There are two known methods for constructing multiple group racks. One is via a $G$-family of racks. The resulting multiple group rack is called the associated multiple group rack of the $G$-family of racks. The other is by taking an abelian extension of a multiple group rack. In this paper, we introduce a new method for constructing multiple group racks by using a $G$-family of racks and a normal subgroup $N$ of $G$. We show that this construction yields multiple group racks that are neither the associated multiple group racks of any $G$-family of racks nor their abelian extensions when the right conjugation action of $G$ on $N$ is nontrivial. As an application, we present a pair of spatial surfaces that cannot be distinguished by invariants derived from the associated multiple group racks of any $G$-family of racks, yet can be distinguished using invariants obtained from a multiple group rack introduced in this paper.

math.GT↗

arXiv2025

A groupoid rack and spatial surfaces

A spatial surface is a compact surface embedded in the $3$-sphere. We assume that a spatial surface is oriented and that each connected component of a spatial surface is neither a disk nor without a boundary. A diagram of a spatial surface is a diagram of a spatial trivalent graph that is a spine of the spatial surface. In this paper, we introduce the notion of a groupoid rack, which is used for considering colorings for diagrams of spatial surfaces in order to obtain an invariant of spatial surfaces. Furthermore, we show that a groupoid rack has a universal property on colorings for diagrams of spatial surfaces.

math.GT↗