Movie moves for singular link cobordisms in 4-dimensional space (Q2796955)

A singular link is the image of an immersion of a disjoint union of circles in \(\mathbb{R}^3\) such that the singularities are finitely many transverse double points, which can be regarded as the image of an embedding in \(\mathbb{R}^3\) of a 4-valent graph with rigid vertices. In this paper, the author introduces the notion of a singular link cobordism in \(\mathbb{R}^4\). Two singular links are cobordant if one is related to the other by singular link isotopy and a combination of births or deaths of simple unknotted curves, and saddle point transformations. Analogous to knotted surfaces, singular link cobordisms can be studied diagrammatically using movie moves, which are movie descriptions of projections in \(\mathbb{R}^3\) with a fixed generic height function. The main theorem is the following Movie-Move Theorem: Two movies represent isotopic singular link cobordisms if and only if one is related to the other by a finite sequence of 24 types of movie moves MM1--MM24 depicted in Figures 7--9 in the paper or interchanges of the levels of distant critical points. The set of movie moves given here include the 7 types of movie parametrizations of the Roseman moves and the 8 types of the Carter-Saito movie moves for knotted surfaces.