Band diagrams of immersed surfaces in 4-manifolds

Abstract

We study immersed surfaces in smooth 4-manifolds via singular banded unlink diagrams. Such a diagram consists of a singular link with bands inside a Kirby diagram of the ambient 4-manifold, representing a level set of the surface with respect to an associated Morse function. We show that every self-transverse immersed surface in a smooth, orientable, closed 4-manifold can be represented by a singular banded unlink diagram, and that such representations are uniquely determined by the ambient isotopy or equivalence class of the surface up to a set of singular band moves which we define explicitly. By introducing additional finger, Whitney, and cusp diagrammatic moves, we can use these singular band moves to describe homotopies or regular homotopies as well.Using these techniques, we introduce bridge trisections of immersed surfaces in arbitrary trisected 4-manifolds and prove that such bridge trisections exist and are unique up to simple perturbation moves. We additionally give some examples of how singular banded unlink diagrams may be used to perform computations or produce explicit homotopies of surfaces.

Minimal generating set of planar moves for surfaces embedded in the four-space

“…6.2] and also [Mar09] with swapped pre-and post-knot, for the singular point case see [Akb16, sec. 5.2] and also [HKM21]). Remark 3.4.…”

“…In the finite cases, the intersection can be a single point (corresponding to a maximal point or to a minimal point) or a four-valent embedded graph, where each vertex corresponds to a saddle point or to a singular point. For more introductory material on this topic in an embedded case refer to [Kam17], and in the immersed case refer to [HKM21].…”

“…Theorem 3.2. [HKM21] Two surface-links immersed in R 4 are equivalent if and only if, their banded unlink diagrams are related by a combination of moves from the set A.…”

“…It was proved later to be a minimal generating set [Jab20]. It was recently extended by a set of three moves intersection/band slide, intersection/band pass, intersection/band swim by M. Hughes, S. Kim and M. Miller [HKM21] to for a generating set of moves relating equivalent immersed surface-links.…”