Kaestner Brackets

TL;DR

This paper introduces Kaestner brackets, a new quantum invariant for virtual knots and links that generalizes biquandle brackets and includes several classical invariants as special cases, providing stronger detection capabilities.

Contribution

The paper develops Kaestner brackets, a novel class of quantum invariants for virtual knots, extending biquandle brackets and unifying multiple classical invariants.

Findings

Kaestner brackets include classical invariants as special cases.

The new invariant is stronger than classical biquandle brackets for virtual knots.

Examples demonstrate the computation and enhanced detection power.

Abstract

We introduce \textit{Kaestner brackets}, a generalization of biquandle brackets to the case of parity biquandles. This infinite set of quantum enhancements of the biquandle counting invariant for oriented virtual knots and links includes the classical quantum invariants, the quandle and biquandle $2$-cocycle invariants and the classical biquandle brackets as special cases, coinciding with them for oriented classical knots and links but defining generally stronger invariants for oriented virtual knots and links. We provide examples to illustrate the computation of the new invariant and to show that it is stronger than the classical biquandle bracket invariant for virtual knots.