Parity Biquandles

TL;DR

This paper introduces Parity Biquandles, a new algebraic structure generalizing biquandles for virtual knots, which helps in estimating crossing numbers and extends to links, providing new tools for virtual knot theory.

Contribution

It defines Parity Biquandles, explores their relation to existing biquandles, and develops polynomial invariants that bound crossing numbers in virtual knots and links.

Findings

Parity Biquandles include all biquandles as even parity cases.

The z-Parity Alexander Biquandle polynomial bounds crossing numbers.

Extension to links provides bounds on inter-component crossings.

Abstract

We use crossing parity to construct a generalization of biquandles for virtual knots which we call Parity Biquandles. These structures include all biquandles as a standard example referred to as the even parity biquandle. Additionally, we find all Parity Biquandles arising from the Alexander Biquandle and Quaternionic Biquandles. For a particular construction named the z-Parity Alexander Biquandle we show that the associated polynomial yields a lower bound on the number of odd crossings as well as the total number of real crossings and virtual crossings for the virtual knot. Moreover we extend this construction to links to obtain a lower bound on the number of crossings between components of a virtual link.