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\begin{document}

%-------------------------------------------------------------------


\section*{singular knot}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{knot_theory}{}\paragraph*{{Knot theory}}\label{knot_theory}

[[!include knot theory - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{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 \emph{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 \textbf{(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

[[!redirects singular knots]] [[!redirects Singular knot]] [[!redirects Singular knots]]



\end{document}