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I am interested in singularity theory of differential mappings by topology. In knot theory, it is well-known thea two knots are equivalent if and only if one differ the other by some moves (i.e. Reidemeister moves). similar moves exists in the other mathematical objects. For example, surfaces in 4-space (Roseman moves) etc. Now I read a paper for moves for surfaces in 3-space. In the paper, by using stratified Morse theory (Cerf theory), complete moves are determined. I do not understand little for cerf theory. However, I am understanding to be determined complete Reidemeister type moves. The moves was determined by using singularities of a projection of proper isotopy and critical points. In the interval between critical points, not change the topology.

Question

When we have known complete moves as above, how do we understand to make the moves? Maybe, after and before in a critical points change the topology.

Thank you for your considerations.

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I'm guessing you are asking about the explicit isotopy corresponding to the combinatorial move. Usually this is described explicitly in the lead-up to the theorem you're interested in. For example, the Reidemeister type 1 move corresponds to the resolutions of a cusp-type singularity in the planar projection of a knot. One resolution substitutes a regular double-point in the projection for the cusp singularity, the other resolution removes the singularity. The actual isotopy in 3-space in general can be quite complicated, but the simplest situations can be done by rigid motions. –  Ryan Budney Jul 17 '11 at 12:24
    
Thank you for your comments. however, I want to know in detail more. For example, in the cusp singularity, there exist the others two singularties. How the neighborhood of two singularities and the one of cusp singularities connect by isotopy? Aren't these isotopy called "moves"? –  muta yasushi Jul 17 '11 at 13:09
    
What two other singularities are you talking about? With your last question, it seems you're blurring together quite different notions. Isotopy has to do with actual embeddings -- these are not diagrams. In the case of classical knots, these are actual embeddings of the circle into $\mathbb R^3$. Moves are relations between planar projections. You say one planar projection is related to the other by a move. So this is quite different than isotopy. –  Ryan Budney Jul 17 '11 at 13:14
    
Thanks. I considered an isotopy with singularities. That is too bad. I must consider a stable map with singularities as a projection of an isotopy. In classical knot case, we must consider a stable map of a surface into 3-space as a projection of a surface in 4-space. It is obvious that these singularities can not connect by same topology. but in isotopy level, these can connect. –  muta yasushi Jul 17 '11 at 14:48
    
Recently, I have realized the isotopy is knot itself in classical knot case, that is surface knot. The projection is generic. an inverse of regular value to isotopy is a regular point.Therefore, projetion type do not change, that is, not move. –  muta yasushi Jul 22 '11 at 10:39
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