Definition:Knot (Knot Theory)

Given a submanifold $$X \subset Y \ $$ and an inclusion $$i:X \to Y \ $$ such that $$i(X) = X \ $$, a knotted embedding is an embedding $$\phi:X \to Y \ $$ (or the image of such an embedding) such that $$\phi(X) \ $$ is not homotopic to $$i(X) \ $$.

Sphere Knot
Frequently, the term "knotted $$n \ $$-sphere is given to a knotted embedding $$\phi:\mathbb{S}^n \to \mathbb{R}^{n+2} \ $$.

Circle Knot
The description of the sphere is dropped for $$\mathbb{S}^1 \ $$ and the term "knot" is used without qualification for knotted embeddings $$\phi:\mathbb{S}^1 \to \mathbb{R}^3 \ $$.

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 $$\mathbb{R}^3 \ $$ 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.