Link Invariants
Examples
Related concepts
CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A singular knot is a smooth map $S^1 \to \mathbb{R}^3$ 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.