I interested in co-dimension 2 projections of knots. A Knot is a embedded circle in 3-space. We want to project it into 1-space. Then we use a Morse function and it appears critical points as singularities. According to the singularity theory, A knot move is made by a surface knot and a projection. For example, Reidemeister moves are considered as neighborhoods of singularities of surfaces in 3-space. The surfaces in 3-space is considered as a co-dimension 1 projection of a surface knot which is a embedded surface into 4-space and represent a knot isotopy. Similarly, we want to consider a co-domension 2 projection of a surface knot. Then it appears folds and cusps as singularities. The neighborhoods of a fold and a cusp may make moves. For recostructing from a projection into 1-space of a knot, we replace a knot as a set of braids with critical point's information. A Morse functions of a knot make intervals between critical points. The pre-image of the interbal is a braid with critical point's information. To neighborhoods of a fold and a cusp, we add the information of braids with critical point's information, that is maybe new knot moves.


Can we make new knot moves for Morse functions like above? Moreover, is it well-known the new move?

Thank you for your considerations.

  • $\begingroup$ perhaps "abobe" is "above?" $\endgroup$ – Will Jagy Mar 3 '12 at 22:16

The answer to your question is a qualified, yes. The full reference is this article by Cooper, Mond and Wit Atique. In it they describe complex multi-germs of functions. This singularity theory is an ingredient in any approach to the Reidemeister moves for higher dimensional knots (beyond 2-knots in 4-space).

For knotted surfaces, the original work of Roseman and independently Homme/Nagase gives the 7 moves that are necessary to move a knotted surface around. The best movie move version of Roseman's theorem is work of mine with Joachim Rieger and Masahico Saito. The non-pay wall version is here . But see also our book Knotted Surfaces and their diagrams or Surfaces in $4$-space.

To get a good understanding of cusps and how they behave, I suggest this recent book . So having shamelessly promoted my work on this let me describe a little on how to approach the general problem in higher dimensions.

Staring from the Reidemeister moves of an $n$-manifold embedded in $(n+2)$-space (and considering their projection in $(n+1)$-space). We use these to construct the singularities of $(n+1)$-manifolds in $(n+3)$ space. These singularities together with the Morse critical points of the $(n+1)$-manifold are the ingredients used to create the knotting. To determine the Reidemeister moves of $(n+1)$-manifolds in $(n+2)$-space one first posits that the lower dimensional R-moves are invertible on both sides. (So for example a there is a type-II saddle and a type-II bubble move). One posits a high dimensional version of the R-III move. In dim. 4 this is the tetrahedral move. In general it corresponds to moving the hyperplane $\sum_j x_j = 1$ across the coordinate planes $x_j=0$ to the plane $\sum_j x_j=-1$. The remaining moves are going to involve the branch points and the analogues of the R-III move. Specifically, branch points can be moved through transverse sheets. This is where Cooper-Mond-Wit Atique is needed.

Finally, I think one should be able to construct a higher dimensionsal movie move theorem by means of examining the interactions between the lower dimensional R-moves, and the critical points of the various strata. As long as you can determine that singularities are codimension 1 type, then you have all the moves.

  • $\begingroup$ Very thanks! You have been my best teacher on surface knot theory. However, this answer is codimension 1 projection version? As Reidemeister moves, there exists diagrammatic moves for a classical knot by using a Morse function? That is a projected to 1-space classical knot diagram's move. $\endgroup$ – muta yasushi Mar 4 '12 at 3:54
  • $\begingroup$ Well, if I understand your question, then yes the knot moves can be interpreted as Morse singularities of knots in 1-higher dimensions. But there are still details to be worked out. In my answer above, I outlined how I think it works. I'll be in Kyoto in May for ILDT and I can explain more then. $\endgroup$ – Scott Carter Mar 4 '12 at 7:13
  • $\begingroup$ Thanks. If possible, I want to see you. But I am not resercher and bussiness person at Fukuoka city office in Japan. Therefore, it is difficult to see you. For example, it exists an discovered knot invariant via those 1-higher dimension's moves? $\endgroup$ – muta yasushi Mar 4 '12 at 10:20

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