On the Euler characteristics for quandles

TL;DR

This paper introduces a new concept of Euler characteristics for quandles, linking algebraic structures to topological invariants, and computes these characteristics for various finite quandles.

Contribution

It defines Euler characteristics for quandles and demonstrates their properties, showing parallels with topological Euler characteristics and computing them for specific examples.

Findings

Quandle Euler characteristic of symmetric spaces matches topological Euler characteristic.

Calculated Euler characteristics for generalized Alexander, core quandles, and discrete manifolds.

Established properties of quandle Euler characteristics akin to topological invariants.

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.