Braiding link cobordisms and non-ribbon surfaces

Abstract

Abstract. We define the notion of a braided link cobordism in S 3 × [0, 1], which generalizes Viro's closed surface braids in R 4 . We prove that any properly embedded oriented surface W ⊂ S 3 × [0, 1] is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary when ∂W already consists of closed braids. These surfaces are closely related to another notion of surface braiding in D 2 × D 2 , called braided surfaces with caps, which are a generalization of Rudolph's braided surfaces. We mention several applications of braided surfaces with caps, including using them to apply algebraic techniques from braid groups to studying surfaces in 4-space, as well as constructing singular fibrations on smooth 4-manifolds from a given handle decomposition.

Annular Khovanov–Lee homology, braids, and cobordisms

“…As noted in[23, Sec. 2.2], a singular still of a movie presentation of a braided cobordism F will change the diagram by one of:…”

“…Note that Hughes has proved in [23] that every link cobordism between braid closures is isotopic rel boundary to a braided cobordism.…”

“…When (L, o) is the annular closure of a braid σ ∈ B n equipped with its braid-like orientation o ↑ (see Subsection 3.2), we can say more. In particular, the work of Hughes in [23] tells us that every link cobordism is isotopic to a so-called braided cobordism (Definition 11), and this isotopy can be realized rel boundary if the original links are already braided with respect to a common axis. This has the following consequence for the annular Rasmussen invariants: Corollary 3.…”

“…[23] A braided cobordism is a smoothly, properly embedded surfaceF ⊂ S 3 × [0, 1] on which the projection pr 2 : S 3 × [0, 1] → [0, 1] restricts as a Morse function with each regular level set F ∩ (S 3 × {t}) a closed braid in S 3 × {t}.…”

Broken Lefschetz fibrations, branched coverings, and braided surfaces

“…If ∂ S is already a closed braid, then the isotopy can be chosen rel ∂ S. Proof. In [23], the author proves that every such surface S ⊂ D 2 × D 2 is isotopic to a braided surface with caps. Here, a cap is an embedded disk in S on which the projection pr 2 restricts as an embedding, and whose boundary is a fold circle as defined above.…”