Virtual Biquandle Cocycle Quiver Representations

TL;DR

This paper develops new invariants for virtual knots and links using quiver representations and biquandle structures, leading to novel polynomial invariants through decategorification.

Contribution

It introduces a framework for virtual knot invariants based on quiver representations and virtual biquandles, expanding the tools for virtual knot theory.

Findings

Defined new quiver representation-valued invariants for virtual knots.

Constructed infinite families of polynomial invariants.

Demonstrated applications to virtual knot classification.

Abstract

We introduce quiver representation-valued invariants of oriented virtual knots and links associated to a choice of finite virtual biquandle, abelian group, set of virtual Boltzmann weights, commutative unital ring and set of virtual biquandle endomorphisms. As an application we define new infinite families of polynomial virtual knot and link invariants via decategorification.