$N$-degeneracy in rack homology and link invariants

Abstract

AbstractWe consider rack homology for racks with rack rank 1 N < 1. For such racks, we prove thatN-degenerate chains form a sub-complex of the classical complex de ning rack homology. This allowsa homology theory analogous to quandle homology for non-quandle racks with nite ranks. If our rackhas rack rank N = 1 then it is a quandle and the homology theory coincides with the CKJLS homologytheory [7]. We use nontrivial cocycles to de ne invariants of knots and provide examples of calculationsfor knots up to 8 crossings and links up to 7 crossings. Keywords: Finite racks, rack homology, enhancements of counting invariants, cocycle invariants2000 MSC: 57M27, 57M25 1 Introduction Racks are algebraic structures with axioms derived from Reidemeister moves type II and type III. They havebeen considered by knot theorists in order to construct knot and link invariants and their higher analogues(see for example [8] and references therein). Racks allow a re ned and a complete algebraic framework inwhich ones investigates links and 3-manifolds. They have been studied by many authors and appeared inthe literature with di erent names such as automorphic sets and in a special case quandles, distributivegroupoids, crystals etc. Rack cohomology was introduced by Fenn, Rourke and Sanderson [12]. For eachrack X and abelian group A, they de ned cohomology groups H