On the rack homology of graphic quandles

TL;DR

This paper reviews rack homology of self-distributive structures, focuses on graphic quandles, computes their second homology groups for many cases, and proposes conjectures based on computational results.

Contribution

It provides new computations of second rack homology groups for a broad class of graphic quandles and introduces conjectures informed by these findings.

Findings

Computed second rack homology for many graphic quandles

Identified patterns leading to new conjectures

Connected rack homology to knot theory applications

Abstract

This paper has partially a novel and partially a survey character. We start with a short review of rack (two term) homology of self distributive algebraic structures (shelves) and their connections to knot theory. We concentrate on a sub-family of quandles satisfying the graphic axiom. For a large family of graphic quandles (including infinite ones), we compute the second rack homology groups. Finally, we propose conjectures based on our computational data.