nLab singular knot

Idea

A singular knot is a smooth map (or isotopy class of smooth maps) $S^1 \to \mathbb{R}^3$ which is not an embedding. That is, it is not a knot. Singular knots come in various flavours according to the ways in which it is possible to fail to be an embedding. One of the simplest type of singular knot is where the only failures allowed are simple double points. That is, places where the curve intersects itself transversally, and at the intersection only two threads intersect.

Formally, $f \colon S^1 \to \mathbb{R}^3$ has a (simple) double point at $x \in \mathbb{R}^3$ if $f^{-1}(x) = \{t_1, t_2\}$ with $t_1 \ne t_2$ and $\{f'(t_1), f'(t_2)\}$ are linearly independent.

Singular knots are an important piece of the theory of Vassiliev finite type invariants of knots.

category: knot theory