Bridge trisections of knotted surfaces in 𝑆⁴

Abstract

We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of fourmanifolds and extends the classical concept of bridge splittings of links in the threesphere to four dimensions. We prove that every knotted surface in the four-sphere admits a bridge trisection (a decomposition into three simple pieces) and that any two bridge trisections for a fixed surface are related by a sequence of stabilizations and destabilizations. We also introduce a corresponding diagrammatic representation of knotted surfaces and describe a set of moves that suffice to pass between two diagrams for the same surface. Using these decompositions, we define a new complexity measure: the bridge number of a knotted surface. In addition, we classify bridge trisections with low complexity, we relate bridge trisections to the fundamental groups of knotted surface complements, and we prove that there exist knotted surfaces with arbitrarily large bridge number. arXiv:1507.08370v1 [math.GT]

Bridge trisections of knotted surfaces in 4-manifolds

“…In [MZ17a], the authors proved that every knotted surface in S 4 admits a generalized bridge trisection in which the underlying trisection of S 4 is the standard genus zero trisection. We will refer to such a decomposition simply as a bridge trisection.…”

“…Gay and Kirby proved that every smooth, closed, connected, orientable 4-manifold (henceforth, 4-manifold ) X admits a trisection, splitting X into three simple 4-dimensional pieces (4-dimensional 1-handlebodies) that meet pairwise in 3dimensional handlebodies and have as their common intersection a closed surface. Similarly, in [MZ17a] the authors proved that every smoothly embedded, closed surface (henceforth, knotted surface) K in S 4 admits a bridge trisection, a decomposition of the pair (S 4 , K) into three collections of unknotted disks in 4-balls that intersect in trivial tangles in 3balls, akin to classical bridge splittings in S 3 . In this paper, we extend this construction to knotted surfaces in arbitrary 4-manifolds.…”

Trisections of surface complements and the Price twist

“…In Figure 16, we consider S = (spun trefoil)#(unknotted torus) ⊂ S 4 (a triplane diagram can be obtained by connect-summing diagrams from [MZ1]; we convert this into a shadow diagram). The torus S can be isotoped so that the standard (0, 0)-trisection (X 1 , X 2 , X 3 ) of S 4 induces a (2, 6)-bridge trisection of S. We stabilize each X i once along an arc in X j ∩ X k ∩ S to find a (3, 1)-trisection of S 4 inducing a (1, 3)-bridge trisection of S; the shadow diagram (Σ, α, β, γ, s α , s β , s γ ) of Figure 16 illustrates this bridge trisection.…”

“…We depict a shadow diagram for S in Figure 17 (top). This diagram can be obtained by understanding [MZ1] (either the explicit example diagrams of twist-spun knots or the procedure to turn a movie of a knotted surface into a triplane diagram). We obtain a relative trisection T = (Σ , α , β , γ ) of S 4 \ν(S) as in Section 4.1.…”

Determining the Trisection Genus of Orientable and Non-Orientable PL 4-Manifolds through Triangulations

“…An extended set of examples of trisections of 4-manifolds can be found in [14]. The recent works of Gay [15], and Meier, Schirmer and Zupan [17,19,20] give some applications and constructions arising from trisections of 4-manifolds and relate them to other structures on 4-manifolds.…”