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In the classical dimension (knots in $S^3$), it is considered standard (I think?) that the following sets are in bijective correspondence:

  • locally flat knots up to ambient isotopy;
  • PL-knots up to PL ambient isotopy;
  • smooth knots up to smooth ambient isotopy.

This should be "by work of Moise" but I am having trouble finding references.

The textbooks of Burde-Zieschang (Proposition 1.10) and Kawauchi (Appendix A) seem to indicate that the set of PL-knots up to ambient isotopy coincides with the set of PL-knots up to PL-ambient isotopy. Some authors also work with tame knots which are defined as knots that are ambient isotopic to a PL-knot. So, it seems that the previous two sets also agree with the set of tame knots up to either ambient or PL ambient isotopy.

Does someone know where I can find the statements that involve the locally flat and smooth cases?

The closest I could get was 1.11.6 and 1.11.7 of Cromwell's book that seems to sketch a proof that the set of PL-knots up to isotopy coincides with the set of smooth knots up to ambient isotopy (the category mixing is unfortunate). Most textbooks appear to stick to the PL knots.

Related but distinct MO posts include: Reference for a fact (?) on homeomorphic knot complements as well as Reference request: A knot is tame if and only if it has a tubular neighbourhood

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  • $\begingroup$ Have you looked in Lickorish's book? I think some of this may be in there, but I won't have it handy for a few days. $\endgroup$
    – Greg Friedman
    Dec 11, 2021 at 13:03
  • $\begingroup$ I have. On page 2, Lickorish says "An alternative way of avoiding wildness is to require that L be a smooth 1-dimensional submanifold of the smooth 3-manifold S3. That leads to an equivalent theory, but in these low dimensions simplexes are often easier to manipulate than are sophisticated theorems of differential manifolds" There are no references. The only instance of locally flat embeddings occurs during the discussion of slice knots. $\endgroup$
    – Anthony Conway
    Dec 11, 2021 at 15:54

1 Answer 1

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Tame knots are PL was proved by Bing and Moise in 1954.

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  • $\begingroup$ @SamHopkins thanks, fixed. The doi on the jstor page was faulty somehow, I should have checked it first. $\endgroup$
    – Ian Agol
    Feb 17, 2022 at 21:32

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