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Let $Y$ be a manifold and $X \subset Y$ a submanifold of $Y$.

Let $i: X \to Y$ be an inclusion, i.e. a mapping such that $i \left({X}\right) = X$.

Then a knotted embedding is an embedding $\phi: X \to Y$ (or the image of such an embedding) such that $\phi \left({X}\right)$ is not freely homotopic to $i \left({X}\right)$.

Sphere Knot

A knotted $n$-sphere is a knotted embedding:

$\phi: \Bbb S^n \to \R^{n + 2}$

Circle Knot

The description of the sphere is dropped for $\Bbb S^1$ and the term knot is used without qualification for knotted embeddings $\phi: \Bbb S^1 \to \R^3$.

Elementary Knot

Circle knots can often be quite wild and unwieldy - most of modern knot theory concerns itself with a specific kind of knot.

These knots are described as a finite set of points in $\R^3$ called $\left\{{ x_1, x_2, \dots, x_n }\right\}$, together with line segments from $x_i$ to $x_{i+1}$ and a line segment from $x_n$ to $x_1$.

The union of all these line segments is clearly a circle knot, or an unknot, an embedding of the circle which is homotopic to a circle.