Knot

Remarks

Equivalence of knots is an equivalence relation, and knot theory studies the equivalence classes. Since $S^1$ is compact, the map $f$ is a homeomorphism onto $K$. Most texts consider only tame knots. Every knot that is the image of a smooth embedding is tame, and every tame knot is equivalent to such a knot. Some texts use the $3$-sphere $S^3$ in place of $\R^3$; this gives the same theory. Often a knot is also oriented, by choosing a direction along $S^1$.

Example: the circle $\set{(x, y, 0) : x^2 + y^2 = 1}$ is a knot, the image of $t \mapsto (\cos t, \sin t, 0)$, which is injective on $[0, 2\pi)$; it is a tame knot, the unknot. Non-example: the figure-eight curve $t \mapsto (\sin 2t, \sin t, 0)$ is not a knot, because it is not injective: $t = 0$ and $t = \pi$ both map to the origin. A wild knot satisfies the first clause but not the third.