$N$-degeneracy in rack homology and link 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 S. Nelson, Link invariants from finite racks, arXiv:0808.0029. 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. 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.