Singquandles, psyquandles, and singular knots: A survey

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 singquandles, and review recent developments regarding invariants of singular knots obtained from psyquandles.