What is a Singular Knot?

TL;DR

This paper explores the concept of singular knots, their role in Vassiliev invariants, and recent advances in extending knot invariants to singular knots, highlighting their algebraic and quantum algebra connections.

Contribution

It reviews recent results on extending non-numerical knot invariants to singular knots and discusses the algebraic and quantum algebra perspectives.

Findings

Singular knots induce a Vassiliev filtration on knot classes.

Extensions of knot invariants to singular knots have been recently developed.

Connections between knot theory, quantum algebra, and singular knots are elucidated.

Abstract

A singular knot is an immersed circle in $\mathbb R^{3}$ with finitely many transverse double points. The study of singular knots was initially motivated by the study of Vassiliev invariants. Namely, singular knots give rise to a decreasing filtration on the infinite dimensional vector space spanned by isotopy classes of knots: this is called the Vassiliev filtration, and the study of the corresponding associated graded space has lead to many insights in knot theory. The Vassiliev filtration has an alternative, more algebraic definition for many flavours of knot theory, for example braids and tangles, but notably not for knots: this view gives rise to connections between knot theory and quantum algebra. Finally, we review results -- many of them recent -- on extensions of non-numerical knot invariants to singular knots.