Homotopical interpretation of link invariants from finite quandles

TL;DR

This paper links quandle cocycle invariants of links to homotopy theory, revealing their torsion components relate to classical invariants and homology groups of quandle classifying spaces.

Contribution

It provides a homotopical interpretation of quandle invariants, connecting them to Dijkgraaf-Witten invariants and homology groups of classifying spaces.

Findings

$ ext{l}$-torsion of invariants equals sum of coloring polynomial and Dijkgraaf-Witten part

Computed third homology and second homotopy groups of quandle classifying spaces

Established a topological meaning for quandle cocycle invariants

Abstract

This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles $X$, from a perspective of homotopy theory: Specifically, for any prime $\ell$ which does not divide the type of $X$, the $\ell$-torsion of this invariants is equal to a sum of the colouring polynomial and a $\Z$-equivariant part of the Dijkgraaf-Witten invariant of a cyclic branched covering space. Moreover, our homotopical approach involves application of computing some third homology groups and second homotopy groups of the classifying spaces of quandles, from results of group cohomology.