What is a Singular Knot?

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.