Birack shadow modules and their link invariants

TL;DR

This paper introduces a new algebraic structure called birack shadow modules and uses it to enhance link invariants, providing more powerful tools for distinguishing knots and links beyond traditional invariants.

Contribution

The authors define the associative algebra Z[X,S] for birack shadows and develop shadow module enhancements that improve knot and link invariants.

Findings

Shadow module invariants are not determined by Alexander polynomial.

Enhanced invariants distinguish knots that unenhanced invariants cannot.

Examples demonstrate the effectiveness of shadow module enhancements.

Abstract

We introduce an associative algebra Z[X,S] associated to a birack shadow and define enhancements of the birack counting invariant for classical knots and links via representations of Z[X,S] known as shadow modules. We provide examples which demonstrate that the shadow module enhanced invariants are not determined by the Alexander polynomial or the unenhanced birack counting invariants.