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PreprintCC BY 4.0arXiv

Two-numbers and Euler characteristics for quandles

  • Ryoya Kai
  • Akira Kubo
  • Hiroshi Tamaru
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Abstract

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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Cite

BibTeXRIS
BibTeX
@misc{kai2026twonumbers,
  title = {{Two-numbers and Euler characteristics for quandles}},
  author = {Ryoya Kai and Akira Kubo and Hiroshi Tamaru},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05187},
  eprint = {2609.05187},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05187},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Two-numbers and Euler characteristics for quandles
AU  - Ryoya Kai
AU  - Akira Kubo
AU  - Hiroshi Tamaru
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05187
UR  - https://arxiv.org/abs/2609.05187
AB  - 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.
ER  -