Singquandles, Psyquandles and Singular Knots: A Survey

TL;DR

This survey reviews algebraic structures like singquandles and psyquandles used in singular knot theory, highlighting recent invariants and polynomial enhancements that advance understanding of singular knots.

Contribution

It summarizes recent developments in algebraic invariants for singular knots, including new cocycle and polynomial invariants derived from singquandles and psyquandles.

Findings

Development of singquandle cocycle invariants

Introduction of polynomial invariants for singular knots

Advancements in invariants from psyquandles

Abstract

In this short survey we review recent results dealing with algebraic structures (quandles, psyquandles, and singquandles) related to singular knot theory. We first explore the singquandles counting invariant and then consider several recent enhancements to this invariant. These enhancements include a singquandle cocycle invariant and several polynomial invariants of singular knots obtained from the singquandle structure. We then explore psyquandles which can be thought of as generalizations of oriented signquandles, and review recent developments regarding invariants of singular knots obtained from psyquandles.