Parity Biquandle Invariants of Virtual Knots

TL;DR

This paper introduces new invariants for virtual knots based on parity biquandles, which can distinguish knots that previous invariants could not, by using counting and cocycle enhancement methods.

Contribution

It develops parity biquandle invariants combining biquandle 2-cocycles and maps, leading to novel polynomial invariants for virtual knots.

Findings

Parity cocycle invariants distinguish virtual knots beyond non-parity invariants.

Examples demonstrate the effectiveness of the new invariants.

The invariants include one-variable and two-variable polynomial forms.

Abstract

We define counting and cocycle enhancement invariants of virtual knots using parity biquandles. These invariants are determined by pairs consisting of a biquandle 2-cocycle \phi^0 and a map \phi^1 with certain compatibility conditions leading to one-variable or two-variable polynomial invariants of virtual knots. We provide examples to show that the parity cocycle invariants can distinguish virtual knots which are not distinguished by the corresponding non-parity invariants.