Rack module enhancements of counting invariants

Abstract

We introduce a modified rack algebra $\mathbb{Z}[X]$ for racks $X$ with finite rack rank $N$. We use representations of $\mathbb{Z}[X]$ into rings, known as rack modules, to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide computations and examples to show that the new invariants are strictly stronger than the unenhanced counting invariant and are not determined by the Jones or Alexander polynomials.