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.