The shadow quandle cocycle invariant of knotoids

TL;DR

This paper introduces shadow quandle cocycle invariants to distinguish knotoids and their mirrors, demonstrating chirality detection and calculating crossing numbers for complex knotoids.

Contribution

It develops shadow quandle cocycle invariants as a new tool for analyzing knotoids, including their chirality and crossing number computations.

Findings

Shadow quandle cocycle invariant distinguishes infinitely many knotoids from their mirrors.

The knot-type knotoid 3_1 is shown to be chiral.

Crossing numbers of multi-linkoids are calculated using quandle 3-cocycle weights.

Abstract

This paper studies the chirality of knotoids using shadow quandle colorings and the shadow quandle cocycle invariant. The shadow coloring number and the shadow quandle cocycle invariant is shown to distinguish infinitely many knotoids from their mirrors. Specifically, the knot-type knotoid $3_1$ is shown to be chiral. The weight of a quandle 3-cocycle is used to calculate the crossing numbers of infinitely many multi-linkoids.