Polynomial knot and link invariants from the virtual biquandle

TL;DR

This paper introduces a new polynomial invariant for virtual knots and links based on the Alexander biquandle, utilizing Gr"obner bases to unify classical and virtual knot invariants.

Contribution

It develops computable Gr"obner basis invariants from the Alexander biquandle that generalize and unify classical and virtual knot polynomials.

Findings

Gr"obner bases fully determine elementary ideals

Invariants distinguish virtual knots effectively

Examples demonstrate the invariants' usefulness

Abstract

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering $<$, the Gr\"obner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to illustrate the usefulness of these invariants and propose questions for future work.