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.