Polynomial Birack Modules

TL;DR

This paper introduces a new enhancement of the birack counting invariant using modules over Laurent polynomial rings, leading to a stronger invariant that captures more information than previous invariants.

Contribution

It develops birack modules over Laurent polynomial rings to define a new polynomial-style invariant that enhances the birack counting invariant.

Findings

The new invariant is stronger than the unenhanced birack counting invariant.

It is not determined by the generalized Alexander polynomial.

Examples demonstrate the effectiveness of the new invariant.

Abstract

Birack modules are modules over an algebra Z[X] associated to a finite birack X. In previous work, birack module structures on Z mod n were used to enhance the birack counting invariant. In this paper, we use birack modules over Laurent polynomial rings Z_n[q,1/q] to enhance the birack counting invariant, defining a customized Alexander polynomial-style signature for each X-labeled diagram; the multiset of these polynomials is an enhancement of the birack counting invariant. We provide examples to demonstrate that the new invariant is stronger than the unenhanced birack counting invariant and is not determined by the generalized Alexander polynomial.