Knots and their Morse functions

Question

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.

Question

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

Thank you for your considerations.

Answer 1

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.