Quandle Cohomology Quiver Representations

TL;DR

This paper introduces a new family of invariants for classical and virtual knots and links using quandle cohomology quiver representations, leading to four novel polynomial invariants and their generalizations.

Contribution

It develops a framework for quandle cohomology quiver representations to define new polynomial invariants of knots and links, including generalizations to biquandles.

Findings

Defined quiver representation-valued invariants for knots and links.

Constructed four new polynomial invariants from these representations.

Extended the framework to biquandles and provided example computations.

Abstract

We define a family of quiver representation-valued invariants of oriented classical and virtual knots and links associated to a choice of finite quandle $X$, abelian group $A$, set of quandle 2-cocycles $C\subset H^2_Q(x;A)$, choice of coefficient ring $k$ and set of quandle endomorphisms $S\subset \mathrm{Hom}(X,X)$. From this representation we define four new polynomial (or ``polynomial'' depending on $A$) invariants. We generalize to the case of biquandles and compute some examples.