**Definitions:**

- A
*knot*is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal). - Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are
*equivalent*if one of the following equivalent conditions is satisfied:- They are ambient isotopic, i. e. there is an isotopy $h:S^3\times I\rightarrow S^3$ with $h_0=\operatorname{id}$, $h_1\circ\kappa=\lambda$.
- There is an orientation preserving homeomorphism $h:S^3\rightarrow S^3$ such that $h\circ\kappa=\lambda$.

**Claim:**
Let $\kappa:S^1\hookrightarrow S^3$ be a knot.
The following are equivalent:

- $\kappa$ is equivalent to a polygonal knot.
- $\kappa$ can be lifted to an embedding of a full torus $\overline{\kappa}:S^1\times D^2\hookrightarrow S^3$.

This claim appears without proof in the following places:

- Louis H. Kauffman:
*Knots and Functional Integration,*Ch. 1.1 (p. 5). homepages.math.uic.edu/~kauffman/KFI.pdfUnless otherwise specified I shall deal only with

*tame*knots and links. In a tame knot every point on the knot has a neighborhood in 3-space that is equivalent to the standard (ball, arc-diameter) pair. Tame knots (links) can be represented up to ambient isotopy by*piecewise linear*knots and links. A link is piecewise linear if the embedding consists in straight line segments. Thus a piecewise linear link is an embedding of a collection of boundaries of $n$-gons (different $n$ for different components and the $n$’s are not fixed). A piecewise linear knot (link) is made from “straight sticks.” - Wikipedia (en):
*Wild knot.*en.wikipedia.org/wiki/Wild_knotIn the mathematical theory of knots, a knot is

*tame*if it can be “thickened up”, that is, if there exists an extension to an embedding of the solid torus $S^1\times D^2$ into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain.

Introduction to Smooth Manifolds(which also suffices for my need). $\endgroup$ – Lilalas Jul 11 at 19:33