On the Euler characteristics for quandles

Ryoya Kai; Hiroshi Tamaru

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arXiv · 2411.08319

On the Euler characteristics for quandles

Ryoya Kai·Hiroshi Tamaru

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Abstract

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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Ryoya Kai, Hiroshi Tamaru. 2024-11-13. On the Euler characteristics for quandles. https://arxiv.org/abs/2411.08319

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