Quandle Coloring Quivers of Surface-Links

TL;DR

This paper extends quandle coloring quivers, previously used for knots and links, to oriented surface-links, providing computational examples and defining a new polynomial invariant called the in-degree quiver polynomial.

Contribution

It introduces quandle coloring quiver invariants for surface-links and computes examples for all with ch-index up to 10, expanding the applicability of these invariants.

Findings

Defined quandle coloring quiver invariants for surface-links

Computed examples for all surface-links with ch-index up to 10

Established the in-degree quiver polynomial as a new invariant

Abstract

Quandle coloring quivers are directed graph-valued invariants of oriented knots and links, defined using a choice of finite quandle $X$ and set $S\subset\mathrm{Hom}(X,X)$ of endomorphisms. From a quandle coloring quiver, a polynomial knot invariant known as the \textit{in-degree quiver polynomial} is defined. We consider quandle coloring quiver invariants for oriented surface-links, represented by marked graph diagrams. We provide example computations for all oriented surface-links with ch-index up to 10 for choices of quandles and endomorphisms.