Planar Equivalence of Knotoids and Quandle Invariants

TL;DR

This paper explores the classification of planar knotoids using specialized quandle invariants, introducing new tools that detect planarity and analyze symmetry properties, including a stronger cocycle invariant.

Contribution

It introduces a novel triangular quandle cocycle invariant and demonstrates its effectiveness over existing invariants for planar knotoids.

Findings

Quandle colorings can detect planarity of knotoids.

Triangular quandle cocycle invariant is stronger than end-specific colorings.

Invariants help analyze symmetry properties like invertibility and chirality.

Abstract

While knotoids on the sphere are well-understood by a variety of invariants, knotoids on the plane have proven more subtle to classify due to their multitude over knotoids on the sphere and a lack of invariants that detect a diagram's planar nature. In this paper, we investigate equivalence of planar knotoids using quandle colorings and cocycle invariants. These quandle invariants are able to detect planarity by considering quandle colorings that are restricted at distinguished points in the diagram, namely the endpoints and the point-at-infinity. After defining these invariants we consider their applications to symmetry properties of planar knotoids such as invertibility and chirality. Furthermore we introduce an invariant called the triangular quandle cocycle invariant and show that it is a stronger invariant than the end specified quandle colorings.