TITLE: Reference request: A knot is tame if and only if it has a tubular neighbourhood
QUESTION [9 upvotes]: 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.pdf

Unless 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_knot

In 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.

REPLY [4 votes]: The existence of PL tubular neighborhoods, plus a uniqueness statement, may be found in Wall, Locally flat PL submanifolds with codimension two, (Proc. Cambridge Philos. Soc. 63 (1967), 5–8.). Is that what you are looking for?