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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:
    1. They are ambient isotopic, i. e. there is an isotopy $h:S^3\times I\rightarrow S^3$ with $h_0=\operatorname{id}$, $h_1\circ\kappa=\lambda$.
    2. 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:

  1. $\kappa$ is equivalent to a polygonal knot.
  2. $\kappa$ 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:

  1. 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.”

  2. 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.

Danny Ruberman
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Lilalas
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    That's basically the PL tubular neighbourhood theorem, together with the observation that the normal bundle is trivial. Triviality of the normal bundle follows from Poincare duality. Probably the best place to look for the theorem written up the way you would like would be a knot theory book that uses the PL category. Perhaps Burde and Ziechang's book? Apologies, I don't have it with me right now. – Ryan Budney Jul 11 '19 at 07:56
  • @RyanBudney If by “trivial” normal bundle you mean isomorphic to $S^1\times D^2$, then the Tubular Neighbourhood Theorem gives me the direction 1⇒2. How do I obtain the other direction? – Lilalas Jul 11 '19 at 19:32
  • I could not find anything useful in Burde-Zieschang either; a smooth version of the Tubular Neighbourhood Theorem is written down in John M. Lee’s Introduction to Smooth Manifolds (which also suffices for my need). – Lilalas Jul 11 '19 at 19:33
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    @Lilalas For the other direction the only reference I know is Moise's book on Geometric Topology in dimensions 2 and 3. – Moishe Kohan Jul 12 '19 at 00:00
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    @MoisheKohan Good idea but what exact statement are you referring to? The closest thing I could find in Moise is Thm. 24.9 -- just after he introduces the notion of a “combinatorial solid torus”. I don’t think that’s exactly what I am looking for, though. – Lilalas Jul 12 '19 at 16:47
  • I tried to look as well, but couldn't find it in Moise. – horned-sphere Oct 21 '21 at 00:10

1 Answers1

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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?

Danny Ruberman
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