# Is there a combinatorial version of PL ambient isotopy in dimension $>3$?

The classical PL Reidemeister Theorem reads:

Reidemeister Theorem: Two knots in $S^3$ are PL ambient isotopic if and only if any diagram of one can be transformed into a diagram of the other by Reidemeister moves.

Recall that PL ambient isotopy of $K_1\colon\, S^1\hookrightarrow S^3$ to $K_2\colon\, S^1\hookrightarrow S^3$ is an orientation-preserving PL map from $S^{3}\times I$ to $S^3$, whose restriction to a map from $S^{3}\times \{t\}$ to $S^3$ is a PL homeomorphism for every $t\in I$, whose restriction to $S^{3}\times \{0\}$ is the identity map, and whose restriction to $S^{3}\times \{1\}$ composed with $K_1$ is $K_2$.

However, every source that I know (see here) proves an a-priori weaker version of the theorem, in which "PL ambient isotopy" is replaced by "combinatorial isotopy" which means: "related by a finite sequence of triangle moves"- replacing 2 (or 1) edges of a triangle which does not intersect the knot in its interior with the other edge (edges). See e.g. these notes by J. Roberts. Indeed, this is the version of the Reidemeister Theorem which Reidemeister himself proved.

I think that the seldom-cited but often-used fact that PL ambient isotopy coincides with combinatorial isotopy of knots in $S^3$ is proved (in the more general case of links, which extends also to knotted graphs as pointed out by Kauffman) in:

Graeub, W. (1950). Die semilinearen abbildungen (pp. 3-70). Springer Berlin Heidelberg.

Question 1: Does there exist a proof of this fact in English? I have trouble with the German.
Question 2: Is there a combinatorial version of PL ambient isotopy in higher dimensions? Specifically, I am interested in the case of a mixed 1- and 2-dimensional polyhedron embedded in a PL 4-ball, so that the only non-manifold points are at endpoints of broken lines, where they meet other broken lines or broken planes. The obvious candidate is triangle moves, analogous "tetrahedron moves", and a "mixed dimension move" to move a neighbourhood of the intersection of a line with a plane, which is a bit harder to write down. Is the proof of this to be found anywhere, and is the general version known and shown? (I'm fairly confident it's not in Graeub's book, but perhaps it follows trivially somehow).
• Probably in defining "ambient isotopy" you want to specify that the associated map $S^3\times I\to S^3\times I$ is an isomorphism (or equivalently that for every $t\in I$ the map gives an isomorphism $S^3\times t\to S^3$). – Tom Goodwillie Nov 27 '13 at 4:21
• Totally different comment: You mention "mixed 1- and 2-dimensional polyhedron inside a PL 4-ball". I don't know what you mean exactly, but for a 1-dimensional polyhedron in a 3-ball that is not a manifold how would you move a non-manifold point by triangle moves? – Tom Goodwillie Nov 27 '13 at 4:38
• @TomGoodwillie : Comment 1: Corrected- thanks! Comment 2: By a "triangle move" in which one edge of the triangle is in a broken line and another is in the broken plane, leaving the edge in the 2-cell in place and choosing the 3rd edge instead of the second edge. Each broken line and each broken surface should be embedded, so the non-manifold points I'm allowing are only where two such objects meet, which has to be at endpoints of the broken lines (I'm also allowing many broken lines to meet at one point). I've edited the question to make this explicit- thanks! – Daniel Moskovich Nov 27 '13 at 5:34

## 1 Answer

Unfortunately, this is not quite an answer. So for knotted surfaces in 4-space, there is Roseman's Theorem. I think that the context of Dennis's proof is in the smooth category. Or certainly the proof that I know depends on a smooth structure. I would imagine that the proof could be cranked up to PL-locally flat, but I have my hesitations about non-simple branch points. Roseman outlines the higher dimensional case as well, and this outline follows the classification of multi-germs, or actually real pictures of complex multi-germs. If you give me N, and a large enough amount of time, I think that I can work out all the moves by hand, but again in a smooth category.

So fundamentally the idea is to take a codimension 2 embedding, and project it into (n+1)-space. Do so generically. Now examine the critical points and intersections of an isotopy. You can sort of build the whole theory up inductively. Your R-moves and critical events in dimension N become the singularities in dimension N+1. My paper on RR moves for foams on the arxiv can be used to fill in the details in the smooth case.

• This was sort-of what I was thinking if it's not already worked-out somewhere. I have been looking at your 1993 "movie moves" paper with Saito, and at Roseman's "Reidemeister-type moves" paper. It just looks kind-of cumbersome in mixed dimension... I'll look into it further though. Thanks! – Daniel Moskovich Nov 27 '13 at 5:28
• A locally flat PL proof of Roseman's Theorem could be a huge help, if one exists; even more so for foams. – Daniel Moskovich Nov 27 '13 at 7:09
• Upon closer investigation, this answer turns out to be more helpful than I originally realized... thanks for the answer! – Daniel Moskovich Dec 1 '13 at 10:24