N-Degeneracy in rack homology and link invariants

TL;DR

This paper introduces a new homology theory for racks with finite rank N, generalizing existing invariants and providing computational examples for classical knots and links.

Contribution

It defines a homology theory for racks of finite rank N, extending quandle homology and knot invariants, with explicit calculations for small knots and links.

Findings

N-degenerate chains form a sub-complex

Homology coincides with CKJLS for rack rank 1

Nontrivial cocycles produce new knot invariants

Abstract

The aim of this paper is to define a homology theory for racks with finite rank N and use it to define invariants of knots generalizing the CJKLS 2-cocycle invariants related to the invariants defined in [15]. For this purpose, we prove that N -degenerate chains form a sub-complex of the classical complex defining rack homology. If a rack has rack rank N = 1 then it is a quandle and our homology theory coincides with the CKJLS homology theory [6]. Nontrivial cocycles are used to define invariants of knots and examples of calculations for classical knots with up to 8 crossings and classical links with up to 7 crossings are provided.