Theorem 1.

Let X be a 4-manifold with trisection T. Any knotted surface K in X can be isotoped into bridge position with respect to T.

If K⊂X is in bridge position with respect to a trisection, we call the decomposition (X,K)=(X1,K∩X1)∪(X2,K∩X2)∪(X3,K∩X3) a generalized bridge trisection.

Returning to dimension three, we note that it can be fruitful to modify a bridge splitting of a knot K in a 3-manifold Y so that the complexity of the underlying Heegaard splitting increases while the number of unknotted arcs decreases. This process involves a technical operation called meridional stabilization. We show that there is an analogous operation, which we also call meridional stabilization, in the context of bridge trisections. As a result, we prove the next theorem. (Precise definitions are included in Section 1.) A 2-knot is a knotted surface homeomorphic to S2.

Theorem 2.

Let K be a knotted surface with n connected components in a 4-manifold X. The pair (X,K) admits a (g,k;b,n)–generalized bridge trisection satisfying b=3⁢n−χ⁢(K). In particular, if K is a 2-knot in X, then K can be put in 1-bridge position.

A generalized bridge trisection of the type guaranteed by Theorem 2 is called efficient with respect to the underlying trisection T, since it is the smallest possible b and n for any surface with the same Euler characteristic.

As a corollary to Theorem 1, we explain how these decompositions provide a way to encode a knotted surface combinatorially in a 2D diagram, which we call a shadow diagram. We anticipate that this paradigm in the study of knotted surfaces will open a window to structures and connections in this field.

Corollary 3.

Every generalized bridge trisection of a knotted surface K in a 4-manifold X induces a shadow diagram. Moreover, if K has n components, then an efficient generalized bridge trisection of K induces a shadow diagram with 9n−3χ(K) arcs. In particular, if K is a 2-knot in X, then (X,K) admits a doubly-pointed trisection diagram.

A knot that has been decomposed into a pair unknotted arcs admits a representation called a doubly pointed Heegaard diagram; a doubly-pointed trisection diagram is a direct adaptation of this structure. See Section 2 for further details.

In Section 2, we give shadow diagrams for various examples of simple surfaces in 4-manifolds. First, we give a classification of those 2-knots that can be put in 1-bridge position with respect to a genus one trisection of the ambient 4-manifold. We also study complex curves in complex 4-manifolds, announcing preliminary results related to ongoing work with Peter Lambert-Cole. In particular, we announce the following result, which shows that complex curves in ℂℙ2 have efficient generalized bridge trisections with respect to the genus one trisection of ℂℙ2.

Theorem 4.

Let Cd be the complex curve of degree d in ℂ⁢ℙ2. Then, the pair (ℂ⁢ℙ2,Cd) admits an efficient generalized bridge trisection of genus one.

This theorem can be used to prove the existence of efficient exotic trisections, which in this setting are defined to be (g,0)–trisections of 4-manifolds that are homeomorphic but not diffeomorphic to a standard 4-manifold.

Section 3 contains the proofs of the main theorems and corollaries. In Section 4, we turn our attention to the question of uniqueness of generalized bridge trisections. To this end, we offer the following conjecture.

Conjecture 5.

Any two generalized bridge trisections for a pair (X,K) that induce isotopic trisections of X can be made isotopic after a sequence of elementary perturbation and unperturbation moves.

Lemma 8.

Let V be a 1-handlebody, and let L be an unlink contained in ∂V. Up to an isotopy fixing L, the unlink L bounds a unique collection of trivial disks in V.

Proof:

It is well-known that the statement is true when V is a 4-ball (3). Suppose that D and D′ are two collections of trivial disks properly embedded in V such that ∂D=∂D′=L. Let B denote a collection of properly embedded 3-balls in V cutting V into a 4-ball, and let S=∂B. Since L is an unlink, we may isotope S in ∂V (along with B in V) so that L∩S=∅. Since D is a collection of boundary parallel disks in V, there exists a set of disks D∗⊂∂V isotopic to D via an isotopy fixing L. Choose D∗ so that the number of components of D∗∩S is minimal among all such sets of disks.

We claim that D∗∩S=∅. First, we observe that every embedded 2-sphere S⊂∂V bounds a properly embedded 3-ball in V: If S does not bound a 3-ball in ∂V, then either S is an essential separating sphere, splitting ∂V into two components, each of which is a connected sum of copies of S1×S2, or S is an essential nonseparating sphere, and there is an S1×S2 summand of ∂V in which S is isotopic to {pt}×S2. In either case, we can cap off ∂V with 4-dimensional 3-handles and a 4-handle to obtain a 1-handlebody in which S bounds a 3-ball. However, this capping-off process is unique (4), and thus S bounds a 3-ball in V as well.

To prove the claim, suppose by way of contradiction that D∗∩S≠∅, and choose a curve c of D∗∩S that is innermost in a sphere component of S, so c bounds a disk E in this component such that int(E)∩D∗=∅. Note that c also bounds a subdisk D of a component of D∗, so S=E∪D is a 2-sphere embedded in ∂V. By the above argument, S bounds a 3-ball B that is properly embedded in V; thus, int(B)∩D∗=∅. It follows that there is an isotopy of D∗ through B in V that pushes D onto E. If D∗′ is the set of disks obtained from D∗ by removing D and gluing on a copy of E (pushed slightly off of S), then D∗′ is isotopic to D∗, with |D∗′∩S|<|D∗∩S|, a contradiction.

It follows that D∗∩S=∅, and we conclude that after isotopy D is contained in the 4-ball W=V∖ν(B). Similarly, we can assume that after isotopy D′ is contained in W. It now follows from ref. 3 that D and D′ are isotopic, as desired. □

Corollary 9.

A generalized bridge trisection is uniquely determined by its spine.

We also observe that we can compute the Euler characteristic of a surface K from the parameters of a generalized bridge trisection. If K⊂X is in bridge position with respect to a trisection T of X, we will set the convention that ci=|K∩Xi| and b=|K∩Hij|=|K∩Σ|/2. The next lemma follows from a standard argument in ref. 1.

Lemma 10.

Suppose that K⊂X is in bridge position with respect to a trisection T. Thenχ(K)=c1+c2+c3−b.

As with trisections, we may wish to assign a complexity to generalized bridge trisections. The most specific designation has eight parameters. If T is a generalized bridge trisection, and the underlying trisection has complexity (g;k1,k2,k3), we say that the complexity of the generalized bridge trisection is (g;k1,k2,k3;b;c1,c2,c3). In the case that k=k1=k2=k3 and c=c1=c2=c3, we say that T is balanced and denote its complexity by (g,k,b,c). Even more generally, a (g,b)-generalized bridge trisection refers to a generalized bridge trisection with a genus g central surface that meets K in 2b points. In Section 2, we classify all (1,1)-generalized bridge trisections. If the underlying trisection is the genus zero trisection of S4, as in ref. 1, we call T a (b;c1,c2,c3)-bridge trisection or a (b,c)-bridge trisection in the balanced case.

A. Shadow Diagrams

Just as a trisection diagram determines the spine of a trisection, a type of diagram called a triplane diagram determines the spine of a bridge trisection, as shown in ref. 1. Unfortunately, triplane diagrams do not naturally extend from bridge trisections to generalized bridge trisections. Instead, we use a structure called a “shadow diagram.” Let τ be a trivial tangle in a handlebody H. A curve-and-arc system (α,a) determining (H,τ) is a collection of pairwise disjoint simple closed curves α and arcs a in Σ=∂H such that α determines H and a is a collection of shadow arcs for τ. Note that curves in α and arcs in a can be chosen to be disjoint by standard cut-and-paste arguments using compressing disks for H and bridge disks for τ. A shadow diagram for a generalized bridge trisection T is a triple ((α,a),(β,b),(γ,c)) of curve-and-arc systems determining the spine (H31,τ31)∪(H12,τ12)∪(H23,τ23) of T. Since every trivial tangle in a handlebody can be defined by a curve-and-arc system, it is clear that Corollary 3 follows immediately from Theorem 1.

B. The 1-Bridge Trisections

One family which deserves special consideration is the collection of 1-bridge trisections, i.e., (g,1)-generalized bridge trisections. If K has a such a splitting, then it intersects each sector Xi of the underlying trisection in a single disk and each handlebody in a single arc. In this case, we deduce from Lemma 10 that K is a 2-knot, and the generalized bridge trisection is efficient. Shadow diagrams for generalized bridge trisections of this type are particularly simple: A doubly-pointed trisection diagram is a shadow diagram in which each curve-and-arc system contains exactly one arc. In this case, drawing the arc in the diagram is redundant, since there is a unique way (up to admissible slides of the arcs and curves) to connect the two points in the complement of any one of the sets of curves.

In Fig. 1, we depict several doubly pointed diagrams for low-complexity examples. First, we give diagrams for the two simplest complex curves in ℂℙ2, namely, the line ℂℙ1 and the quadric C2. Next, we give diagrams for an S2–fiber in S2×S2 and the sphere C(1,1) in S2×∼S2 representing (1,1)∈ℤ⊕ℤ≅H2(S2×∼S2). (See ref. 5 for formal definitions.) We postpone the justification for these diagrams until Section 3, in which we develop the machinery to make such justification possible.

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Fig. 1.

Some doubly-pointed trisection diagrams. From left to right: (ℂℙ2,ℂℙ1), (ℂℙ2,C2), (S2×S2,S2×{∗}), and (S2×∼S2,C(1,1)).

By results in refs. 1, 6, and 7, any surface with a (0,b)–generalized bridge trisection (i.e., a b-bridge trisection) for b<4 is unknotted in S4. In Section 3, we will prove the following classification result.

Proposition 11.

There are exactly two nontrivial (1,1)–knots (up to change of orientation and mirroring): (ℂ⁢ℙ2,ℂ⁢ℙ1) and (ℂ⁢ℙ2,C2).

On the other hand, there are many 2-knots admitting (3,1)-generalized bridge trisections: Perform three meridional stabilizations (defined in Section 3) on any (4,2)-bridge trisection, of which there are infinitely many (1). We offer the following as worthwhile problems.

Problem 12.

Classify 2-knots admitting (2,1)-generalized bridge trisections and projective planes admitting (1,2)-generalized bridge trisections.

With regard to Problem 12, Fig. 2 shows a (2,1)–shadow diagram for the standard projective (real) plane (ℂℙ2,ℝℙ2) that is the lift of the standard cross-cap in S4 with normal Euler number −2 under the branched double covering. More generally, consider a surface knot (or link) (X,K), and let Xn(K) denote the n-fold cover of X, branched along K. Let K∼n denote the lift of K under this covering.

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Fig. 2.

The branched double covering projection relating the standard cross-cap (S4,ℙ+) and its cover (ℂℙ2,ℝℙ2).

Proposition 13.

If (X,K) admits a (g;k1,k2,k3;b;c1,c2,c3)–generalized bridge trisection, then (Xn⁢(K),K∼) admits a (g′;k1′,k2′,k3′;b,c1,c2,c3)–generalized bridge trisection, where g′=n⁢g+(n−1)⁢(b−1) and ki′=n⁢ki+(n−1)⁢(ci−1).

Proof:

It is a standard exercise to show that the n-fold cover of a genus g handlebody branched along a collection of b trivial arcs is a handlebody of genus g′=ng+(n−1)(b−1), with the lift of the original b trivial arcs being a collection of b trivial arcs upstairs. From this, the rest of the proposition follows, once we observe that the trivial disk system (♮k(S1×B3),D) is simply the trivial tangle product (♮k(S1×D2),τ)×I, and that the branched covering respects this product structure. Thus, each piece of the trisection lifts to a standard piece, so the cover is trisected. □

C. Complex Curves in ℂℙ2.

In this subsection, we summarize results that have been obtained in collaboration with Peter Lambert-Cole regarding generalized bridge trisections of complex curves in complex 4-manifolds of low trisection genus (e.g., ℂℙ2, S2×S2, and ℂℙ2#ℂℙ2¯). Let Cd denote the complex curve of degree d in ℂℙ2. Note that Cd is a closed surface of genus (d−1)(d−2)/2.

Theorem 4.

The pair (ℂ⁢ℙ2,Cd) admits a (1,1;(d−1)⁢(d−2)+1,1)-generalized bridge trisection.

In other words, complex curves in ℂℙ2 admit efficient generalized bridge trisections with respect to the genus one trisection of ℂℙ2; each such curve can be decomposed as the union of three disks (c=1). See Fig. 3. Let Xn,d denote the 4-manifold obtained as the n-fold cover of ℂℙ2, branched along Cd, which exists whenever n divides d. When n=d, we have that Xd,d is the degree d hypersurface in ℂℙ3. The next corollary follows from Theorem 4 and Proposition 13.

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Fig. 3.

Two shadow diagrams for C3 in ℂℙ2. The diagram on the left is due to Peter Lambert-Cole, and the diagram on the right is efficient.

Corollary 14.

Xn,d admits an efficient (g,0)-trisection where g=n+(n−1)⁢(d−1)⁢(d−2).

Note that Zp,q,r=pℂℙ2#qℂℙ2¯#rS2×S2 admits as (g,0)-trisection where g=p+q+2r. It had been speculated that an extension of the main theorems of refs. 6 and 7 would show that every manifold admitting a (g,0)-trisection is diffeomorphic to Zp,q,r; however, Corollary 14 gives many interesting counterexamples to this suspicion.

For example, if d is odd and at least five, then Xd,d is homeomorphic, but not diffeomorphic, to Zp,q,0 for certain p,q≥0 (8). Thus, we see that there are pairs of exotic manifolds that are not distinguished by their trisection invariants. We note that Baykur and Saeki have previously given examples of inefficient exotic trisections (9).

A. The Existence of Generalized Bridge Splittings.

Here, we discuss the interaction between handle decompositions and trisections of closed 4-manifolds. We will not rigorously define handle decompositions but direct the interested reader to ref. 5.

Suppose H is a handle decomposition of a 4-manifold X with a single 0-handle and a single 4-handle. Corresponding to H, there is a Morse function h:X→ℝ, equipped with a gradient-like vector field that induces the handle decomposition H. We will suppose that each Morse function is equipped with a gradient-like vector field (which we will neglect to mention henceforth). After an isotopy, we may assume that every critical point of index i occurs in the level h−1(i). Such a Morse function is called self-indexing. For any subset S⊂ℝ, let YS denote X∩h−1(S). Let Z be a compact submanifold of Y{t} for some t, and let [r,s] be an interval containing t. We will let Z[r,s] denote the subset of X obtained by pushing Z along the flow of h during time [r,s]. (For example, if t∈(r,s), this will involve pushing Z up and down the flow.) In particular, if this set does not contain a critical point of h, then Z[r,s] is diffeomorphic to Z×[r,s]. We let Z{t′} denote Z[r,s]∩h−1(t′).

Now, let H be a handle decomposition of X with ni i-handles for i=1,2,3, and let T be the attaching link for the 2-handles, so that T is an n2-component framed link contained in #n1(S1×S2) with Dehn surgery yielding #n3(S1×S2). In addition, let h:X→ℝ be a self-indexing Morse function inducing H. We suppose without loss of generality that T is contained in Y{3/2}=#n1(S1×S2), and we let Σ be a genus g Heegaard surface cutting Y{3/2} into handlebodies H and H′, where a core of the handlebody H contains T.

The following lemma is essentially lemma 14 of ref. 2, in which it is proved in slightly different terms.

Lemma 15.

Let X be a 4-manifold with self-indexing Morse function h, and, using the notation above, consider the setsX1=Y[0,3/2]∪H′[3/2,2],X2=H[3/2,5/2],X3=H′[2,5/2]∪Y[5/2,4].The decomposition X=X1∪X2∪X3 is a (g;n1,g−n2,n3)-trisection with central surface Σ{2}.

Given a self-indexing Morse function h and surface Σ as above, we will let T(h,Σ) denote the trisection described by Lemma 15. The next lemma also comes from ref. 2; it is a restatement of lemma 13 from that work.

Lemma 16.

Given a trisection T of X, there is a self-indexing Morse function h and surface Σ⊂Y{3/2} such that T=T⁢(h,Σ).

Now we turn our focus to knotted surfaces in 4-manifolds. Suppose that K is a knotted surface in X. A Morse function of the pair h:(X,K)→ℝ is a Morse function h:X→ℝ with the property that the restriction hK is also Morse. Note that for any K⊂X, a Morse function h:X→ℝ becomes a Morse function of the pair (X,K) after a slight perturbation of K in X. Expanding upon the previous notation, for S⊂ℝ, we let LS denote K∩h−1(S). Let J be a compact submanifold of L{t} for some t, and let [r,s] be an interval containing t. We will let J[r,s] denote the subset of K obtained by pushing J along the flow of hK during time [s,r]. As above, if this set does not contain a critical point of hK, then J[r,s] is diffeomorphic to J×[r,s].

Saddle points of hK can be described as cobordisms between links obtained by resolving bands: Given a link L in a 3-manifold Y, a band is an embedded rectangle R=I×I such that R∩L=∂I×I. We resolve the band R to get a new link by removing the arcs ∂I×I from L and replacing them with the arcs I×∂I. Note that every band R can be represented by a framed arc η=I×{1/2}, so η meets L only in its endpoints. Let h be a Morse function of the pair (X,K), suppose that all critical points of h and hK occur at distinct levels, and let x∈K be a saddle point contained in the level h−1(t). Then, there is a framed arc η with endpoints in the link L{t−ϵ} with the property that the link L{t+ϵ} is obtained from L{t−ϵ} by resolving the band corresponding to η. We will use this fact in the proof of the next lemma, which is related to the notion of a normal form for a 2-knot in S4 (10, 11).

Lemma 17.

Suppose X is a 4-manifold equipped with a handle decomposition H, and K is a surface embedded in X. After an isotopy of K, there exists a Morse function of the pair (X,K) such that h is a self-indexing Morse function inducing the handle decomposition H, and index i critical points of hK occur in the level Y{i+1}.

Proof:

Let Γ1 be an embedded wedge of circles containing the cores of the 1-handles, so that ν(Γ1) is the union of the 0-handle and the 1-handles of H. Similarly, let Γ3 be an embedded wedge of circles such that ν(Γ3) is the union of the 3-handles and 4-handle. After isotopy K meets Γ1 and Γ3 transversely; hence K∩Γ1=K∩Γ3=∅, and thus we can initially choose a self-indexing Morse function h:X→ℝ so that ν(Γ1)=Y[0,1+ϵ), ν(Γ3)=Y(3−ϵ,4], and K⊂Y(1+ϵ,3−ϵ). For each minimum point of hK, choose a descending arc avoiding K and the critical points of h and drag the minimum downward within a neighborhood of this arc until it is contained in Y{1}. Similarly, there is an isotopy of K after which all maxima are contained in Y{3}.

It only remains to show that after isotopy, all saddles of hK are contained in Y{2}. Let T be the attaching link for the 2-handles of H, considered as a link in Y{2}. For each saddle point xi in level ti<2, let ηi be the framed arc with endpoints in L{ti}, where 1≤i≤n, so that Lti+ϵ is obtained from Lti−ϵ by resolving the band induced by ηi. Certainly, η1 is disjoint from Lt1−ϵ except at its endpoints. A priori, η2 may intersect the band induced by η1, but after a small isotopy, we may assume that η2 avoids η1 and thus we can push η2 into Y{t1}. Continuing this process, we may push all arcs ηi into Y{t1}, and generically, the graph Lt1−ϵ∪{ηi} is disjoint from T, so the entire apparatus can be pushed into Y{2}. A parallel argument shows that the framed arcs coming from saddles occurring between t=2 and t=3 can be pushed down into Y{2}, as desired. □

We call a Morse function h:(X,K)→ℝ that satisfies the conditions in Lemma 17 a self-indexing Morse function of the pair (X,K). Given such a function, we can push the framed arcs {ηi} corresponding to the saddles of hK into the level Y{3/2}, where the endpoints of {ηi} are contained in L{3/2} and resolving L{3/2} along the bands given by {ηi} yields the link L{5/2}. A banded link diagram for K consists of the union of L{3/2} with the bands given by {ηi}, contained in Y{3/2}, along with the framed attaching link for the 2-handles in X, denoted by T⊂Y{3/2}. As such, a banded link diagram completely determines the knotted surface K⊂X. Let H be the handle decomposition of X determined by h. As above, let Σ be a Heegaard surface cutting Y{3/2} into handlebodies H and H′, where a core of H contains T.

Let Γ=L{3/2}∪{ηi} in Y{3/2}. We will show that Γ may be isotoped to be in a relatively nice position with respect to the surface Σ, from which it will follow that there is an isotopy of K to be in a relatively nice position with respect to the trisection T(h,Σ). An arc η⊂∂H is dual to a trivial arc τi⊂H if there is a shadow τi′ for τi that meets η in one endpoint. Finally, a collection of pairwise disjoint arcs {ηi}⊂∂H is said to be dual to a trivial tangle {τi} if there is a collection of shadows {τ′i} that meet {ηi} only in their endpoints and such that each component of {ηi}∪{τ′i} is simply connected (in other words, this collection contains only arcs, not loops).

We say that Γ is in bridge position with respect to Σ if the link L{3/2} is in bridge positions and in addition, {ηi}⊂Σ with framing given by the surface framing and the arcs {ηi} are dual to the trivial arcs L{3/2}∩H. To clarify, the arc ηi⊂Σ has framing given by the surface framing exactly when the band induced by ηi meets Σ in the single arc ηi. Next, we show that such structures exist, after which we describe how they induce generalized bridge trisections of (X,K).

Lemma 18.

Given a knotted surface K⊂X and a self-indexing Morse function h of the pair (X,K), let Σ, H, H′, T, and Γ be as defined above. There exists an isotopy of Γ in Y{3/2} after which Γ is in bridge position with respect to Σ.

Proof:

This decomposition is similar to the notion of a banded bridge splitting from ref. 1, where the detailed arguments in theorem 1.3 do not make use of the fact that Σ is sphere and thus transfer directly to this setting. We give a brief outline of the proof, but refer the reader to ref. 1 for further details.

Consider cores C⊂H and C′⊂H′, which may be chosen so that T⊂C and both C and C′ are disjoint from Γ. Note that Y{3/2}∖(C∪C′) is diffeomorphic to Σ×(−1,1) and thus there is a natural projection from Y{3/2}∖(C∪C′) onto Σ=Σ×{0}. By equipping this projection with crossing information, we may view it as an isotopy of Γ within Y{3/2}∖(C∪C′). First, if the arcs {ηi} project to arcs that cross themselves or each other, we may stretch L{3/2} and shrink {ηi} so these crossings are slid to L{3/2}, after which the projection of the collection {ηi} is embedded in Σ. (see figure 10 of ref. 1). It may be possible that some surface framing of some arc ηi disagrees with its given framing; in this case, an isotopy of L{3/2} allows ηi to be pushed off of and back onto Σ with the desired framing, as in figure 11 of ref. 1. Thus, we may assume condition (2) of the definition of bridge position of Γ is satisfied.

Now, we push the projection L{3/2} off of Σ so that L{3/2} is in bridge position, fulfilling condition (1) of the definition of bridge position. At this point, it may not be the case that the arcs {ηi} are dual to L{3/2}∩H; however, this requirement may be achieved by perturbing L{3/2} near the endpoints of the arcs {ηi} in Σ, as in figure 12 of ref. 1. □

Lemma 19.

Suppose that K is a knotted surface in X, with self-indexing Morse function h of the pair (X,K) and Σ, H, H′, T, and Γ as defined above. Suppose further that Γ is in bridge position with respect to Σ, push the arcs {ηi} slightly into the interior of H. Let X1, X2, and X3 be defined as in Lemma 15, and define Di=K∩Xi. Then(X,K)=(X1,D1)∪(X2,D2)∪(X3,D3)is a generalized bridge trisection of (X,K).

Proof:

By Lemma 15, the underlying decomposition X=X1∪X2∪X3 is a trisection, and thus we must show that Di is a trivial disk system in Xi and Di∩Dj is a trivial tangle in the handlebody Xi∩Xj.

Let τ=L{3/2}∩H and τ′=L{3/2}∩H′, so that each of τ and τ′ is a trivial tangle in H and H′, respectively. We note that by construction, D1=L[1,3/2]∪τ′[3/2,2], D2=τ[3/2,5/2], and D3=L[5/2,3]∪τ′[2,5/2]. Thus, there is a Morse function of the pair (X1,D1) that contains only minimal, so that D1 is a collection of trivial disks in X1. Similarly, (X3,D3) contains only maxima, so that D3⊂X3 is a collection of trivial disks as well. We also note that D1∩D3=τ′{2}, a collection of trivial arcs in X1∩X3, and D1∩D2=τ{3/2}∪(∂τ)[3/2,2], a collection of trivial arcs in X1∩X2.

It only remains to show that D2 is a collection of trivial disks in X2, and D2∩D3 is a collection of trivial arcs in X2∩X3. However, this follows immediately from lemma 3.1 of ref. 1; although the proof of lemma 3.1 is carried out in the context of the standard trisection of S4, it can be applied verbatim here. □

Proof of Theorem 1:

By Lemma 16, there exists a self-indexing Morse function h:X→ℝ and Heegaard surface Σ⊂Y{3/2} such that T=T(h,Σ). Applying Lemma 17, we have that there is an isotopy of K after which h:(X,K)→ℝ is a self-indexing Morse function of the pair. Moreover, by Lemma 18, there is a further isotopy of K after which the graph Γ induced by the saddle points of hK is in bridge position with respect to Σ. Finally, the decomposition defined in Lemma 19 is a generalized bridge trisection of (X,K), completing the proof. □

We note that as in lemma 3.3 and remark 3.4 from ref. 1, this process is reversible; in other words, every bridge trisection of (X,K) can be used to extract a handle decomposition of K within X. The proof of lemma 3.3 applies directly in this case, and when we combine it with Lemma 16 above, we have the following:

Proposition 20.

If T is a (g;k1,k2,k3;b;c1,c2,c3)-generalized bridge trisection of (X,K), then there is a Morse function h of the pair (X,K) such that h has k1 index one critical points, g−k2 index two critical points, and k3 index three critical points; and hK has c1 minima, b−c2 saddles, and c3 maxima.

We can now justify the diagrams in Fig. 1. By Proposition 20, a 1-bridge trisection will give rise to a banded link diagram without bands corresponding to a Morse function h of the pair (X,K) such that hK has a single minimum and maximum. From the shadow diagrams in Fig. 1, we extract banded link diagrams, shown directly beneath each shadow diagram. In each case, the black curve in Fig. 1 bounds a disk (the minimum of hK) in the 4-dimensional 0-handle and a disk (the maximum of hK) in the union of the 2-handles with the 4-handle. For example, in the first and third figure, we see that the 2-knot is the union of a trivial disk in the 0-handle, together with a cocore of a 2-handle. The second figure is a well-known description of the quadric. See subsection above. The fourth figure can be obtained by connected summing the first figure with its mirror.

B. Classification of (1,1)-Generalized Bridge Trisections.

In this subsection, we prove Proposition 11, classifying (1,1)-generalized bridge trisections.

Proof of Proposition 11:

Suppose that (X,K) admits a (1,1)-generalized bridge trisection T. Then c1=c2=c3=1, χ(K)=2, and K is a 2-sphere. In addition, by Proposition 20, there is a self-indexing Morse function h on (X,K) so that hK has one minimum, one maximum, and no saddles. If h has no index two critical points, then (X,K) is the double of a trivial disk in a 4-ball or 1-handlebody; thus, K is unknotted. If any one ki=1, then after permuting indices, we may assume that the induced h has no index two critical points. Thus, the only remaining case is k1=k2=k3=0, and so X=ℂℙ2 or ℂℙ¯2.

We will only consider the case X=ℂℙ2; parallel arguments apply by reversing orientations. Let h be a self-indexing Morse function for T, so that Y{3/2} is diffeomorphic to S3, L{3/2} is an unknot we call C, and T is a (+1)-framed unknot disjoint from C in Y{3/2}. In addition, attaching a 2-handle to T yields another copy of S3, in which C remains unknotted. In other words, C is an unknot in S3 that is still unknotted after (+1)-Dehn surgery on T. There are three obvious links C∪T that satisfy these requirements: a two-component unlink, a Hopf link, and the torus link T(2,2). The first of these three corresponds to the unknotted 2-sphere. The next two correspond to ℂℙ1 and C2, respectively. We claim no other links C∪T of this type exist.

Consider T as a (nontrivial) knot in the solid torus S3∖ν(C). Since C remains unknotted after (+1)-surgery on T, it follows that T is a knot in a solid torus with a solid torus surgery. Let ω denote the linking number of C and T, so that ω is also the winding number of T in S3∖ν(C). By ref. 12, one of the following holds: ω=1 and T∪C is the Hopf link, ω=2 and T∪C is the torus link T(2,2), or ω≥3 and the slope of the surgery on T is at least four. The third case contradicts the assumption that the surgery slope is one, completing the proof. □

Remark 21:

A similar argument invoking (13) can be used to show that the only nontrivial 2-knots in S4, ℂ⁢ℙ2, ℂ⁢ℙ¯2, or S1×S3 admitting a (2,1)-generalized bridge trisection are ℂℙ1 and C2, as above.

C. Meridional Stabilization.

Consider a link L in a 3-manifold Y, equipped with a (g,b)-bridge splitting (Y,L)=(H1,τ1)∪(H2,τ2), where b≥2. Fix a trivial arc τ′∈τ2, and let H1′=H1∪ν(τ′)¯ and H2′=H2∖ν(τ′). In addition, let τi′=L∩Hi′, so that τ1′=τ1∪τ′ and τ2′=τ2∖τ′. Then the decomposition (Y,L)=(H1′,τ1′)∪(H2′,τ2′) is a (g+1,b−1)-bridge splitting which is called a meridional stabilization of the given (g,b)-splitting. (See ref. 14, for example.)

In this subsection, we will extend meridional stabilization to a similar construction involving generalized bridge trisections to prove Theorem 2. Let T be a generalized bridge trisection for a connected knotted surface K⊂X with complexity (g;k1,k2,k3;b;c1,c2,c3), and assume that c1≥2. Since K is connected, there exists an arc τ′∈τ23 with the property that the two endpoints of τ′ lie in different components of D1. Define (X1′,D1′)=(X1∪ν(τ′)¯,D1∪(ν(τ′)¯∩K)) and (Xj′,Dj′)=(Xj∖ν(τ′),Dj∖ν(τ′)) for j=2,3, and let T′ be the decomposition(X,K)=(X1′,D1′)∪(X2′,D2′)∪(X3′,D3′).We say that the decomposition T′ is obtained from T via meridional 1-stabilization along τ′. We define meridional i-stabilization similarly for i=2 or 3. Observe that the assumption that K is connected is slightly stronger than necessary; the existence of the arc τ′∈τjk connecting two disks in Di is necessary and sufficient. Notably, T′ is a generalized bridge splitting for (X,K), which we verify in the next lemma. Fig. 4 shows the local picture of a meridional 1–stabilization.

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Fig. 4.

A sample meridional 1-stabilization along τ′ (light green, top right). Meridional stabilization increases the genus of the central surface by one, and a new compressing curve is shown for each handlebody in the bottom half of the figure.

Lemma 22.

The decomposition T′ of (X,K) is a generalized bridge trisection of complexity (g+1;k1+1,k2,k3;b−1;c1−1,c2,c3).

Proof:

Since τ′⊂∂Xj for j=2 and 3, we have that (Xj′,Dj′)≅(Xj,Dj). Let X′=ν(τ′)¯∩(X2∪X3) and D′=ν(τ′)¯∩(D2∪D3). Then X′ is a topological 4-ball intersecting X1 in two 3-balls in ∂X1; i.e., X′ is a 1-handle. It follows that X1′ is obtained from X1 by the attaching a 1-handle, so X1′≅♮k1+1(S1×B3). Similarly, D′ is a band connecting disks D1 and D2 in D1. Since these disks are trivial, we can assume without loss of generality that D1 and D2 have been isotoped to lie in ∂X1, and since D′ is boundary parallel inside X′, the disk D′=D1∪D′∪D2 is boundary parallel in X1′. It follows that D1′=D1∖(D1∪D2)∪D′ is a trivial (c1−1)–disk system.

It remains to verify that the 3D components of the new construction are trivial tangles in handlebodies. Observe that for {j,k}={2,3}, the decomposition (∂Xj,∂Dj)=(H1j′,τ1j′)∪(Hjk′,τjk′) is a 3D meridional stabilization of (∂Xj,∂Dj)=(H1j,τ1j)∪(Hjk,τjk). Thus, τij′ is a trivial (b−1)-strand tangle in the genus g+1 handlebody Hij′, as desired. □

We can now prove Theorem 2, which implies Corollary 3 as an immediate consequence.

Proof of Theorem 2:

Start with a generalized bridge trisection of (X,K). If there is a spanning arc τ′ of the type that is necessary and sufficient for a meridional stabilization, then we perform the stabilization. Thus, we assume there are no such spanning arcs. If Di contains ci disks, then since there are no τ′–type arcs in τjk for i,j,k distinct, it follows that the ci disks belong to distinct connected components of K. Thus, ci=n, and χ(K)=c1+c2+c3−b=3n−b, so that b=3n−χ(K). □

Proposition 23.

Every bridge splitting of an unlink L in #k(S1×S2) is isotopic to some number of perturbations and stabilizations performed on the standard bridge splitting.

Consider a bridge trisection T for a knotted surface K⊂X, with components notated as above. Proposition 23 implies the key fact that K admits a shadow diagram ((α,a),(β,b),(γ,c)) such that a pair of collections of arcs, say a and b for convenience, do not meet in their interiors, and in addition, the union a∪b cuts out a collection of embedded disks D∗ from the central surface Σ. Choose a single component D∗ of these disks together with an embedded arc δ∗ in D∗ which connects an arc a′∈a to an arc b′∈b. Note that D∗ is a trivialization of the disks D1⊂X1 bounded by τ31∪τ12 in ∂X1=H31∪H12, so that we may consider δ∗ and D∗ to be embedded in the surface K. In addition, there is an isotopy of D∗ in ∂X1 pushing the shadows a∪b onto arcs in τ31∪τ12, making D∗ transverse to Σ and carrying δ∗ to an embedded arc in ∂X1 that meets the central surface Σ in one point.

Let Δ be a rectangular neighborhood of δ∗ in D∗, and consider the isotopy of K, supported in Δ, which pushes δ∗⊂K away from X1 in the direction normal to ∂X1. Let K′ be the resulting embedding, which is isotopic to K. The next lemma follows from the proof of lemma 6.1 in ref. 1.

Lemma 24.

The embedding K′ is in (b+1)-bridge position with respect to the trisection X=X1∪X2∪X3, and if ci′=|K′∩Xi|, then c1′=c1+1, c2′=c2, and c3′=c3.

We call the resulting bridge trisection an elementary perturbation of T, and if T′ is the result of some number of elementary perturbations performed on T, we call T′ a perturbation of T. Work in ref. 1 also makes clear how to perturb via a shadow diagram. View the rectangle Δ as being contained in Σ, and parameterize it as Δ=δ∗×I. Now, crush Δ to a single arc c′=∗×I that meets δ∗ transversely once. Considering the arc c′ as a shadow arc for the third tangle, the result is a shadow diagram for the elementary perturbation of T. See Fig. 5.

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Fig. 5.

An illustration (at the level of the shadow diagram) of an elementary 1-perturbation of a generalized bridge splitting.

In ref. 1, the authors prove that any two bridge trisections for a knotted surface (S4,K) are related by a sequence of perturbations and unperturbations. In the setting of generalized bridge trisections, we have the following conjecture.

Conjecture 25.

Any two generalized bridge trisections for (X,K) with the same underlying trisection for X become isotopic after a finite sequence of perturbations and unperturbations.

The proof of the analogous result for bridge trisections in ref. 1 requires a result of Swenton (20) and Kearton-Kurlin (21) that states that every one-parameter family of Morse functions of the pair ht:(S4,K)→ℝ such that ht:S4→ℝ is the standard height function can be made suitably generic. Unfortunately, a more general result does not yet exist for arbitrary pairs (X,K); however, we remark that Conjecture 25 would follow from such a result together with an adaptation of the proof in ref. 1.