Quandle Module Quivers

TL;DR

This paper introduces a new two-variable polynomial invariant for knots and links by enhancing the quandle coloring quiver invariant with quandle modules, which can distinguish knots that previous invariants cannot.

Contribution

It develops a novel enhancement of the quandle coloring quiver invariant using quandle modules, resulting in a more powerful invariant that generalizes previous invariants.

Findings

The new invariant specializes to the previous quandle module polynomial invariant.

It also reduces to the quandle counting invariant.

Example computations demonstrate the enhanced invariant distinguishes more knots and links.

Abstract

We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant with specializes to the previous quandle module polynomial invariant as well as to the quandle counting invariant. We provide example computations to show that the enhancement is proper in the sense that it distinguishes knots and links with the same quandle module polynomial.