arXiv subjects

Ryoya Kai

Publications and source records attributed to Ryoya Kai.

6 recordsLinked to original sources

arXiv2026

Two-numbers and Euler characteristics for quandles

Quandles can be regarded as generalizations of symmetric spaces. In the theory of symmetric spaces developed by Chen and Nagano, there is an interesting relationship between the two-number and the Euler characteristic. The two-number is a Riemannian geometric invariant that can also be characterized in terms of point symmetries, whereas the Euler characteristic is a topological invariant. The aim of this paper is to initiate the study of quandle analogues of Chen--Nagano theory. In particular, we investigate relationships between the two-numbers and the Euler characteristics of quandles, and provide examples of finite quandles that either satisfy or fail to satisfy properties analogous to those of symmetric spaces. These examples are constructed from directed simple graphs labeled by abelian groups.

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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.

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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.

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arXiv2025

Metrics for quandles

A quandle is an algebraic system originating in knot theory, which can be regarded as a generalization of the conjugation of groups. This structure naturally defines two subgroups of its automorphism group, which are called the inner automorphism group and the displacement group, and they act on the quandle from the right. For a quandle with such groups being finitely generated, we investigate the graph structures induced from the actions, and induced metric spaces. The graph structures are defined by the notion of the Schreier graph, which is a natural generalization of the Cayley graph for a group. In particular, the metric associated with the displacement group for an important class of quandles, namely, generalized Alexander quandles, is studied in detail. We show that such a metric space is quasi-isometric to the displacement group with a word metric. Finally, we provide some examples quasi-isometric to typical metric spaces.

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arXiv2024

On the Euler characteristics for quandles

A quandle is an algebraic system whose axioms generalize the algebraic structure of the point symmetries of symmetric spaces. In this paper, we give a definition of Euler characteristics for quandles. In particular, the quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic. Additionally, we calculate the Euler characteristics of some finite quandles, including generalized Alexander quandles, core quandles, discrete spheres, and discrete tori. Furthermore, we prove several properties of quandle Euler characteristics, which suggest that they share similar properties with topological Euler characteristics.

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arXiv2024

On the quandles of isometries of the hyperbolic 3-space

A quandle is an algebraic structure whose axioms are related to the Reidemeister moves used in knot theory. In this paper, we investigate the conjugate quandle of the orientation-preserving isometry group $\mathrm{PSL}(2, \mathbb{C})$ of hyperbolic 3-space and its subquandles. We introduce a quandle, denoted by $Q(\Gamma, \gamma)$, associated with a pair $(\Gamma, \gamma)$. Here, $\Gamma$ is a Kleinian group, and $\gamma$ is a non-trivial element of $\Gamma$. This construction can be regarded as a generalization of knot quandles to hyperbolic knots. Moreover, for pairs $(\Gamma, \gamma)$ satisfying certain conditions, we construct the canonical map from $Q(\Gamma, \gamma)$ to the conjugate quandle of $\mathrm{PSL}(2, \mathbb{C})$, which is an injective quandle homomorphism with a discrete image.

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