A Diagrammatic Construction of Third Homology Classes of Knot Quandles

TL;DR

This paper introduces shadow fundamental classes in third quandle homology of knot quandles, linking them to cocycle invariants and demonstrating their universality for prime knots.

Contribution

It constructs shadow fundamental classes in third quandle homology and establishes their role in cocycle invariants and their universality for prime knots.

Findings

Shadow fundamental classes relate to shadow quandle cocycle invariants.

Any third quandle homology class for a prime knot is an image of a shadow fundamental class.

The paper clarifies the connection between different types of quandle cocycle invariants.

Abstract

We construct elements of the third quandle homology groups of knot quandles, which are called the shadow fundamental classes. They play the same roles for the shadow quandle cocycle invariants of knots as the fundamental classes of knot quandles does for the quandle cocycle invariants. As an application of the shadow fundamental classes, we show the relation between the shadow quandle cocycle invariants and the based shadow quandle cocycle invariants. Moreover, we will show, for a prime knot, that any third quandle homology classes are considered as images of the shadow fundamental classes of some links.