nLab
singular knot
Context
Knot theory
Topology
topology (point-set topology , point-free topology )

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

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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.