nLab singular knot



Knot theory


topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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topological homotopy theory


A singular knot is a smooth map (or isotopy class of smooth maps) S 1 3S^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:S 1 3f \colon S^1 \to \mathbb{R}^3 has a (simple) double point at x 3x \in \mathbb{R}^3 if f 1(x)={t 1,t 2}f^{-1}(x) = \{t_1, t_2\} with t 1t 2t_1 \ne t_2 and {f(t 1),f(t 2)}\{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

Last revised on August 27, 2015 at 15:01:27. See the history of this page for a list of all contributions to it.