Characterizing Dehn surgeries on links via trisections

Edited by David Gabai, Princeton University, Princeton, NJ, and approved June 15, 2018 (received for review October 4, 2017)
October 22, 2018
115 (43) 10887-10893

Significance

Dehn surgery is the process in which one cuts out a neighborhood of a knot or a link in 3D space and reglues this neighborhood in a different way to obtain a new 3D space. By viewing this operation as occurring smoothly over a period, there is a natural interpretation of spaces “before” and “after” a Dehn surgery as level sets of a generic function taking a 4D space to the real line. As such, 4D spaces are deeply linked to Dehn surgeries. This paper explores the ways in which trisections of 4-manifolds can be used to answer interesting questions about such surgeries.

Abstract

We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In particular, we propose a program in which trisections could be used to disprove the generalized property R conjecture, including a process that converts the potential counterexamples of Gompf, Scharlemann, and Thompson into genus four trisections of the standard 4-sphere that are unlikely to be standard. We also give an analog of the Casson–Gordon rectangle condition for trisections that obstructs reducibility of a given trisection.
The theory of Dehn surgery on knots has been thoroughly developed over the past 40 y. In general, this research has focused on two major questions: First, which manifolds can be obtained by surgery on a knot in a given manifold Y? Second, given a pair of manifolds Y and Y, for which knots KY, does there exist a surgery to Y? These two questions have contributed to the growth of powerful tools in low-dimensional topology, such as sutured manifold theory, the notion of thin position, and Heegaard Floer homology. For example, over the last 15 y, the Heegaard Floer homology theories of Ozsváth and Szabó have dramatically deepened our collective understanding of Dehn surgeries on knots (for instance, ref. 1).
If we replace the word “knot” with “link” in the preceding paragraph, the situation changes significantly; for example, the classical Lickorish–Wallace theorem asserts that every 3-manifold Y can be obtained by surgery on a link in S3 (2, 3). For the second general question, concerning which links in a given 3-manifold Y yield a surgery to another given 3-manifold Y, we observe the following basic fact: Two framed links that are handleslide equivalent surger to the same 3-manifold (4). Thus, surgery classification of links is necessarily considered up to handleslide equivalence, and tools which rely on the topology of a knot exterior S3\ν(K) are not nearly as useful, since handleslides can significantly alter this topology.
The purpose of this paper is to make clear the significant role of the trisection theory of smooth 4-manifolds in the classification of Dehn surgeries on links, including a program that suggests trisections may be used to disprove the generalized property R conjecture (GPRC), Kirby problem 1.82 (5). The GPRC asserts that every n-component link in S3 with a Dehn surgery to #n(S1×S2) is handleslide equivalent to the n-component zero-framed unlink. We call a link L with such a surgery an R-link. The related stable GPRC asserts that if L is an R-link, then the disjoint union of L and an unlink is handleslide equivalent to an unlink. The GPRC is known to be true when n=1 (6), and the stable GPRC is known to be true in the following special case.

Theorem 1 (7).

If LS3 is an n-component R-link with tunnel number n, then L satisfies the stable GPRC.
Any n-component R-link L can be used to construct a closed 4-manifold XL, where XL has a handle decomposition with a single 0-handle, no 1-handles, n 2-handles, n 3-handles, and a single 4-handle. An elementary argument reveals that XL is a homotopy 4-sphere, and if L is handleslide equivalent to an unlink, then XL is the standard S4. Thus, both the GPRC and stable GPRC imply the smooth 4D Poincaré conjecture (S4PC) for geometrically simply connected 4-manifolds (those that can be built without 1-handles). Yet these conjectures are substantially stronger than this instance of the S4PC, since the GPRC implies that not only that is XL standard, but also that the handle decomposition can be standardized without adding any canceling pairs of handles. (The stable version allows the addition of canceling 2-handle/3-handle pairs, but not canceling 1-handle/2-handle pairs.) Although experts seem divided about the veracity of the S4PC, it is widely believed that the GPRC is false, with the most prominent possible counterexamples appearing in a paper of Gompf, Scharlemann, and Thompson (8), building on work of Akbulut and Kirby (9).
A new tool that has been useful in this context is a trisection of a 4-manifold, introduced by Gay and Kirby (10). A trisection is a decomposition of a 4-manifold X into three simple pieces, a 4-dimensional version of a 3D Heegaard splitting. Elegantly connecting the two theories, Gay and Kirby (10) proved that every smooth 4-manifold admits a trisection, and every pair of trisections for a given 4-manifold has a common stabilization, mirroring the Reidemeister–Singer theorem (11, 12) in dimension three. Unlike Heegaard splittings, however, the stabilization operation of Gay and Kirby can be broken into three separate operations, called unbalanced stabilizations of types 1, 2, and 3 (7). A trisection is said to be standard if it is an unbalanced stabilization of the genus zero trisection of S4, and thus every trisection of S4 becomes standard after some number of stabilizations. Just as trisections were pivotal in the Proof of Theorem 1 above, we have also used trisections to obtain the following Dehn surgery classification result.

Theorem 2 (13).

If LS3 is a two-component link with tunnel number one with an integral surgery to S3, then L is handleslide equivalent to a 0-framed Hopf link or a ±1-framed unlink.
In the present article, we exhibit a program to disprove the GPRC in three steps, of which we complete the first two. The initial step translates the GPRC and the related stable GPRC into statements about trisections of the 4-sphere. In 3. R-Links and Stabilizations we prove the following, postponing rigorous definitions for now.

Theorem 3.

Suppose L is an R-link and Σ is any admissible surface for L.
i)
If L satisfies the GPRC, then T(L,Σ) is 2-standard.
ii)
The link L satisfies the stable GPRC if and only if T(L,Σ) is 2,3-standard.
The second step is contained in 4. Trisecting the Gompf–Scharlemann–Thompson Examples in which we convert the proposed counterexamples of Gompf–Scharlemann–Thompson into trisections (with explicit diagrams). The final, incomplete step in this program is to prove that the trisections constructed in the second step are not 2-standard, which, together with Theorem 3, would imply that the GPRC is false. To accomplish step iii, we must develop machinery to verify that a trisection is nonstandard. To this end, in 5. A Rectangle Condition for Trisection Diagrams we introduce an analog of the Casson–Gordon rectangle condition (14) for trisection diagrams, giving a sufficient condition for a trisection diagram to correspond to an irreducible (nonstandard) trisection.
We encourage the reader to view this article in full color, as a gray-scale rendering of the figures leads to a loss of information.

1. Trisections

All manifolds are connected and orientable, unless otherwise stated. We let ν() refer to an open regular neighborhood in an ambient manifold that should be clear from context. The tunnel number of a link LY is the cardinality of the smallest collection of arcs a with the property that Y\ν(La) is a handlebody. In this case, ν(La) is a Heegaard surface cutting Y\ν(L) into a handlebody and a compression body. A framed link refers to a link with an integer framing on each component.
Let L be a framed link in a 3-manifold Y, and let a be a framed arc connecting two distinct components of L; call them L1 and L2. The framings of L1, L2, and a induce an embedded surface SY, homeomorphic to a pair of pants, such that L1L2a is a core of S. Note that S has three boundary components, two of which are isotopic to L1 and L2. Let L3 denote the third boundary component, with framing induced by S. If L is the framed link (L\L1)L3, we say that L is obtained from L by a handleslide of L1 over L2 along a.
If two links are related by a finite sequence of handleslides, we say they are handleslide equivalent. It is well known that Dehn surgeries on handleslide-equivalent framed links yield homeomorphic 3-manifolds (4). Recall that an R-link is an n-component link in S3 with a Dehn surgery to the manifold #n(S1×S2), which we henceforth denote by Yn. Let Un denote the n-component zero-framed unlink in S3. If an R-link L is handleslide equivalent to Un, we say that L has property R. If the split union LUr is handleslide equivalent to Un+r for some integer r, we say that L has stable property R. With these definitions the GRPC and stable GPRC can be formulated as follows.

(Stable) GPRC.

Every R-link has (stable) property R.
In this section, we explore the relationship between R-links and trisections of the smooth 4-manifolds that can be constructed from these links.
Let X be a smooth, orientable, closed 4-manifold. A (g;k1,k2,k3)-trisection T of X is a decomposition X=X1X2X3 such that
i)
each Xi is a 4D 1-handlebody, ki(S1×B3);
ii)
if ij, then Hij=XiXj is a 3D handlebody, g(S1×D2); and
iii)
the common intersection Σ=X1X2X3 is a closed genus g surface.
The surface Σ is called the trisection surface or central surface, and the parameter g is called the genus of the trisection. The trisection T is called balanced if k1=k2=k3=k, in which case it is called a (g,k) -trisection; otherwise, it is called unbalanced. We call the union H12H23H31 the spine of the trisection. In addition, we observe that Xi=Yki=HijΣHli is a genus g Heegaard splitting. Because there is a unique way to cap off Yki with ki(S1×B3) (15, 16), every trisection is uniquely determined by its spine.
Like Heegaard splittings, trisections can be encoded with diagrams. A cut system for a genus g surface Σ is a collection of g pairwise disjoint simple closed curves that cut Σ into a2g-punctured sphere. A cut system δ is said to define a handlebody Hδ if each curve in δ bounds a disk in Hδ. A triple (α,β,γ) of cut systems is called a (g;k1,k2,k3)-trisection diagram for T if α, β, and γ define the components Hα,Hβ, and Hγ of the spine of T. We set the conventions that Hα=X3X1, Hβ=X1X2, and Hγ=X2X3, which the careful reader may note differ slightly from conventions in ref. 7. With our conventions, (α,β), (β,γ), and (γ,α) are Heegaard diagrams for Yk1, Yk2, and Yk3, respectively. In ref. 10, Gay and Kirby proved that every smooth 4-manifold admits a trisection, and trisection diagrams, modulo handleslides within the three collections of curves, are in one-to-one correspondence with trisections.
Given trisections T and T for 4-manifolds X and X, we can obtain a trisection for X#X by removing a neighborhood of a point in each trisection surface and gluing pairs of components of T and T along the boundary of this neighborhood. The resulting trisection is uniquely determined in this manner; we denote it by T#T. A trisection T is called reducible if T=T#T, where neither T nor T is the genus zero trisection; otherwise, it is called irreducible. Equivalently, T is reducible if there exists an essential separating curve δ in Σ that bounds compressing disks in Hα, Hβ, and Hγ. Such a curve δ represents the intersection of a decomposing 3-sphere with the trisection surface.
In dimension three, stabilization of a Heegaard surface may be viewed as taking the connected sum with the genus one splitting of S3, and a similar structure exists for trisections. Let Si denote the unique genus one trisection of S4 satisfying ki=1. Diagrams for these three trisections are shown in Fig. 1. A trisection T is called i-stabilized if T=T#Si and is simply called stabilized if it is i-stabilized for some i=1,2,3. Two trisections T and T are called stably equivalent if there is a trisection T that is a stabilization of both T and T. Gay and Kirby (10) proved that any two trisections of a fixed 4-manifold are stably equivalent.
Fig. 1.
The three genus one trisections diagrams for S4.
We say that a trisection T of S4 is standard if T can be expressed as the connected sum of genus one trisections Si.

2. Admissible Surfaces

Here we turn our attention to R-links and Dehn surgeries, before connecting these surgeries to the trisections described above. Recall that Yk denotes #k(S1×S2), and an R-link L is a framed n-component link in S3 such that Dehn surgery on L yields Yn. As mentioned above, every R-link L describes a closed 4-manifold XL with a handle decomposition with a single 0-handle, zero 1-handles, n 2-handles, n 3-handles, and a single 4-handle. Thus, XL is a homotopy S4. An admissible Heegaard surface Σ for L is a Heegaard surface cutting S3 into two handlebodies H and H such that a core of H contains L. As such, M=H\ν(L) is a compression body and Σ may be viewed as a Heegaard surface for the link exterior E(L)=S3\ν(L). Let HL be the handlebody that results from Dehn filling M (or performing Dehn surgery on L in H) along the framing of the link L. An admissible pair consists of an R-link together with an admissible Heegaard surface.
For completeness, we also allow the empty link, L=, where L has an empty Dehn filling yielding S3, giving rise to a handle decomposition of S4=X with only a single 0- and 4-handle. An admissible surface Σ for the empty link is a (standard) genus g Heegaard surface for S3. A genus g Heegaard diagram (α,β) for Yk is called standard if αβ contains k curves, and the remaining gk curves occur in pairs that intersect once and are disjoint from other pairs. A trisection diagram is called standard if each pair is a standard Heegaard diagram. Note that a standard trisection of S4 has a standard diagram, since each of its summands Si has such a diagram.

Lemma 4.

Let L be an n-component R-link. Every admissible pair (L,Σ) gives rise to a trisection T(L,Σ) of XL with spine HHHL. If g(Σ)=g, then T(L,Σ) is a (g;0,gn,n)-trisection. Moreover, there is a trisection diagram (α,β,γ) for T(L,Σ) such that
i)
Hα=H, Hβ=H, and Hγ=HL;
ii)
L is a sublink of γ, where γ is viewed as a link framed by Σ in S3=HαHβ; and
iii)
(β,γ) is a standard diagram for Ygn, where βγ=γ\L.

Proof:

This is proved (in slightly different formats) for L in both refs. 7 and 10. If L=, then it follows easily that S4=X has a handle decomposition without 1-, 2-, or 3-handles, H=HL, and HHHL is the spine for the (g;0,g,0)-trisection T(L,Σ) of S4. In this case, there is a diagram such that β=γ, the standard genus g diagram for Yg.
Note that the conventions Hα=H, Hβ=H, and Hγ=HL, in conjunction with our earlier conventions, identify the 0-handle with X1, the trace of the Dehn surgery on Hβ along L with X2, and the union of the 3-handles and the 4-handle with X3.
Lemma 5 connects R-links, standard trisections, and the stable GPRC.

Lemma 5.

Suppose L is an n-component R-link with admissible genus g surface Σ, and T(L,Σ) is a standard trisection of S4. Then L has stable property R.

Proof:

By Lemma 4, the trisection T(L,Σ) has a diagram (α,β,γ) such that (β,γ) is the standard Heegaard diagram for Ygn. Viewing γ as a g-component link in S3=HαHβ, we have that (gn) curves in γ bound disks in Hβ, while the remaining n curves are isotopic to L [and are disjoint from the (gn) disks]. Thus, as a link in S3, we have γ=LUgn.
In addition, the trisection T(L,Σ) is a standard (g;0,gn,n)-trisection of S4 by hypothesis. As such, it must be a connected sum of gn copies of S2 and n copies of S3, and it has a standard diagram, (α,β,γ), where gn curves in γ are also curves in β, and the remaining n curves are also curves in α. Thus, in S3=HαHβ, the curves γ compose a g-component unlink, with surface framing equal to the zero framing on each component. Since (α,β,γ) and (α,β,γ) are trisection diagrams for the same trisection, we have that γ is handleslide equivalent to γ via slides contained in Σ. Therefore, γ and γ are handleslide-equivalent links in S3. We conclude that L has stable property R, as desired. □
As an aside, we note that Theorem 1 can be obtained quickly using Lemma 5 and the classification of (g;0,1,g1)-trisections from ref. 7.

3. R-Links and Stabilizations

To prove Theorem 3, we develop the connection between R-links, their induced trisections, and the three types of stabilizations. First, we must introduce several additional definitions. Let (L1,Σ1) and (L2,Σ2) be two admissible pairs and define the operation * by
(L1,Σ1)*(L2,Σ2)=(L1L2,Σ1#Σ2),
where the connected sum is taken so that L1L2 is not separated by the surface Σ1#Σ2.

Lemma 6.

If (L1,Σ1) and (L2,Σ2) are admissible pairs, then (L,Σ)=(L1,Σ1)*(L2,Σ2) is an admissible pair, and T(L,Σ)=T(L1,Σ1)#T(L2,Σ2).

Proof:

It is clear that the framed link L1L2 has the appropriate surgery, so L is an R-link. Suppose Σi bounds a handlebody Hi with core Ci containing Li. Then there is a core C for H1H2 such that L1L2C1C2C, and thus Σ1#Σ2 is admissible as well. For the second claim, note that the separating curve δ arising from the connected sum Σ=Σ1#Σ2 is a reducing curve for T(L,Σ), splitting it into the trisections T(L1,Σ1) and T(L2,Σ2). □
Let U be a 0-framed unknot in S3, and let ΣU be the genus one splitting of S3 such that one of the solid tori bounded by ΣU contains U as a core. In addition, let Σ be the genus one Heegaard surface for S3, to be paired with the empty link. Note that (U,ΣU) and (,Σ) are admissible pairs.

Lemma 7.

The pairs (,Σ) and (U,ΣU) yield the following trisections:
i)
T(,Σ)=S2.
ii)
T(U,ΣU)=S3.

Proof:

Note that each trisection in question has genus one. The associated trisections T(,Σ) and T(U,ΣU) are (1;0,1,0)- and (1;0,0,1)-trisections, respectively, and thus they must be S2 and S3. □
By combining Lemmas 6 and 7, we obtain the following.

Corollary 8.

Suppose (L,Σ) is an admissible pair, with T=T(L,Σ):
i)
T((L,Σ)*(,Σ)) is the 2-stabilization of T.
ii)
T((L,Σ)*(U,ΣU)) is the 3-stabilization of T.
In addition, if Σ+ is the stabilization of Σ (as a Heegaard surface for Yk), then (L,Σ+)=(L,Σ)*(,Σ).

Remark 9.

Notably absent from Lemma 7 and Corollary 8 is any reference to 1-stabilization. By generalizing the definition of an admissible pair, we can accommodate 1-stabilization in this context; however, 1-stabilizing a trisection T(L,Σ) that arises from an R-link L corresponds to adding a canceling 1-handle/2-handle pair to the induced handle decomposition of XL. This addition takes us away from the setting of R-links, so we have chosen not to adopt this greater generality here.
We say that two trisections T1 and T2 of a 4-manifold X are 2-equivalent if there is a trisection T that is the result of 2-stabilizations performed on both T1 and T2.

Lemma 10.

If Σ1 and Σ2 are two distinct admissible surfaces for an R-link L, then the trisections T(L,Σ1) and T(L,Σ2) are 2-equivalent.

Proof:

Since both Σ1 and Σ2 are Heegaard surfaces for E(L), they have a common stabilization Σ by the Reidemeister–Singer theorem (11, 12). By Lemma 6, the surface Σ is admissible, and by Corollary 8, T(L,Σ) can be obtained by 2-stabilizations of T(L,Σi).
Observe that 2-equivalence is an equivalence relation. Since Lemma 10 implies that every trisection T(L,Σ) coming from a fixed R-link L belongs to the same 2-equivalence class, it follows that L has a well-defined 2-equivalence class, namely, the 2-equivalence class of T(L,Σ) for any admissible surface Σ. If two R-links L1 and L2 give rise to 2-equivalent trisections, we say that L1 and L2 are 2-equivalent.
Suppose that L is an n-component R-link with admissible surface Σ, cutting S3 into HH, and L is isotopic into a core CH as above. As such, there is a collection of n compressing disks D with the property that each disk meets a unique component of L once and misses the other components. We call D a set of dualizing disks. Note that if (α,β,γ) is the trisection diagram for T(L,Σ) guaranteed by Lemma 4, then the n disks bounded by the n curves in β that are not in γ are a set of dualizing disks for L.

Lemma 11.

If R-links L1 and L2 are related by a handleslide, then L1 and L2 are 2-equivalent.

Proof:

If Li is an n-component link, then L1 and L2 have n1 components in common and differ by a single component, L1L1 and L2L2, where a slide of L1 over another component L of L1 along a framed arc a yields L2. Consider Γ=L1a, an embedded graph with n1 components, and let Σ be a Heegaard surface cutting S3 into HH, where Γ is contained in a core of H. Then L1 is also contained in a core of H, and Σ is admissible (with respect to L1). Let D1 be a set of dualizing disks for L1, which by construction may be chosen so that the arc a avoids all of the disks D1 (Fig. 2).
Fig. 2.
The disks and arcs used in the Proof of Lemma 11, in which the pairs (L1,D1) and (L,D) are replaced with (L2,D1) and (L,D2).
There is an isotopy taking Γ into Σ, preserving the intersections of Li with the dualizing disks D1, so that the framing of Γ agrees with its surface framing in Σ. As such, we can perform the handleslide of L1 over L along a within the surface Σ, so that the resulting link L2 is also contained in Σ, with framing given by the surface framing. Let D1D1 be the disk that meets L1 once, and let DD1 be the disk that meets L once. There is an arc a, isotopic in Σ to an arc in Γ, that connects D1 to D (Fig. 2). Let D2 be the compressing disk obtained by banding D1 to D along a. Then D2=(D1\D)D2 is a set of dualizing disks for L2. Thus, by pushing L2 back into H, we see that Σ is an admissible surface for L2.
Following Lemma 4, let HiHiHLi be a spine for T(Li,Σ). By construction, H1=H2 and H1=H2. Finally, since Hi is Dehn surgery on Li in Hi, and L1 and L2 are related by a single handleslide, we have HL1=HL2. It follows that T(L1,Σ)=T(L2,Σ), and we conclude that L1 and L2 are 2-equivalent.□
Recall that a standard trisection of S4 is the connected sum of copies of S1, S2, and S3 and Un is the zero-framed, n-component unlink, so XUn=S4.

Lemma 12.

Let Σ be any admissible surface for Un; then T(Un,Σ) is standard.

Proof:

We induct on (n,g) with the dictionary ordering. If n=1, then E(U1) is a solid torus. If g=1, then Σ=ΣU, so that T(U1,ΣU)=S3 by Lemma 7. If n=1 and g>1, then Σ is stabilized (17, 18), which means that T(U1,Σ) is 2-stabilized by Corollary 8, and, as such, T(U1,Σ) is standard by induction.
In general, note that the Heegaard genus of an n-component unlink is n; thus gn for all possible pairs (n,g). For n>1, we have that E(Un) is reducible, and so Haken’s lemma (19) implies that Σ is reducible, splitting into the connected sum of genus g1 and g2 surfaces Σ1 and Σ2, where Σi is a Heegaard surface for E(Uni). Then T(Un,Σ)=T(Un1,Σ1)#T(Un2,Σ2), where (ni,gi)<(n,g). Since both summands are standard trisections by induction, it follows that T(Un,Σ) is also standard, completing the Proof. □
A trisection T is said to be 2-standard if it becomes standard after some number of 2-stabilizations. Similarly, T is 2,3-standard if it becomes standard after some number of 2- and 3-stabilizations.

Proof of Theorem 3:

Suppose L has property R. By Lemma 11, L and Un are 2-equivalent links. Thus, T(L,Σ) is 2-equivalent to some trisection coming from Un, but all trisections induced by Un are standard by Lemma 12, and thus T(L,Σ) becomes standard after a finite sequence of 2-stabilizations.
If L has stable property R, then LUn has property R for some n, and thus T((L,Σ)*(U,ΣU)**(U,ΣU)) is 2-standard by the above arguments. By Lemma 7 and Corollary 8,
T((L,Σ)*(U,ΣU)**(U,ΣU))=T(L,Σ)#S3##S3;
hence T(L,Σ) is 2,3-standard.
Finally, if the trisection T(L,Σ) is 2,3-standard, then there exist integers s and t such that the connected sum of T(L,Σ) with s copies of S2 and t copies of S3 is standard. Let (L*,Σ*) be the admissible pair given by
(L*,Σ*)=(L,Σ)*(,Σ)**(,Σ)s×*(U,ΣU)**(U,ΣU)t.
By assumption, T(L*,Σ*) is standard, so by Lemma 5, the link L* has stable property R. But by definition of *, we have L*=LUt, and thus L also has stable property R, completing the Proof. □

4. Trisecting the Gompf–Scharlemann–Thompson Examples

To use Theorem 3 to disprove the GPRC or the stable GPRC, we must convert the possible counterexamples to these theorems into trisections. In this section, we find admissible surfaces related to the examples proposed by Gompf, Scharlemann, and Thompson (8). We call this family the Gompf–Scharlemann–Thompson (GST) links. First, we outline that construction, and then we define the GST links and discuss how they fit into the broader picture. To proceed, we need several new definitions.
Let K be a knot in S3. We say that K is ribbon if K bounds an immersed disk in S3 whose double points are ribbon singularities. It is well known that every ribbon disk can be viewed as a properly embedded disk in the standard 4-ball B4, where KS3=B4. Suppose B is any homotopy 4-ball. The knot K is called homotopy ribbon in B if there exists a properly embedded disk DB such that K=DS3=B and the inclusion map (S3,K)(B,D) induces a surjection π1(S3\K)π1(B\D). Every ribbon knot is homotopy ribbon.
Let KS3 with F a Seifert surface for K. The knot K is fibered with fiber F if its exterior E(K) is homeomorphic to the mapping torus of a homeomorphism φ:FF such that φF=id, called the monodromy of K. Let Y^ denote the 3-manifold obtained by 0-surgery on K in S3. Then Y^ can be constructed by capping off each copy of F with a disk in the fibration of E(K) to get a closed surface F^, so that Y^ is the mapping torus of φ^:F^F^. We call φ^ the closed monodromy of K. Finally, we say that φ extends over a handlebody H if there is a homeomorphism Φ:HH such that ΦH=φ^.
Casson and Gordon (20) proved a remarkable theorem connecting homotopy-ribbon knots to handlebody extensions.

Theorem 13.

Let KS3 be a fibered knot with fiber F and monodromy φ. Then K is homotopy ribbon in a homotopy 4-ball B if and only if the monodromy φ extends over a handlebody H.
As above, let KS3 be a fibered ribbon knot with fiber F, so that Theorem 13 implies that the monodromy φ of K extends over a handlebody H. Let LF be a link in S3 such that L is the boundary of a cut system defining H. We call L a Casson–Gordon derivative of K.

Proposition 14.

Suppose K is a fibered ribbon knot with genus g fiber F and Casson–Gordon derivative L. Then both L and KL are R-links. Moreover, L has a genus 2g admissible surface, and thus the 4-manifold XL admits a (2g;0,g,g)-trisection.
In the remainder of this section, we spell out the details for the simplest case, (p,q)=(3,2). Let Q denote the square knot T(3,2)#T(3,2), let F denote its genus two fiber surface, and let φ denote the monodromy of E(Q). In ref. 21, Scharlemann depicted an elegant way to think about the monodromy φ: We may draw F as a topological annulus A, such that
an open disk D has been removed from A,
each component of A is split into six edges and six vertices, and
opposite inside edges of A are identified, and opposite outside edges of A are identified, so that the quotient space is homeomorphic to F.
With respect to A, the monodromy φ is a 1/6th clockwise rotation of A, followed by an isotopy of D returning it to its original position. As above, let Y^ be the closed 3-manifold obtained by 0-surgery on Q, and let φ^ denote the closed monodromy of Q. Note that φ^ is an honest 1/6th rotation of the annulus in Fig. 3, since, in this case, the puncture has been filled in by the Dehn surgery. Details can be found in refs. 8 and 21, where Lemma 15 is proved.
Fig. 3.
The curves V3/7, V3/7, and V3/7 on the genus two fiber F for the square knot.

Lemma 15.

For every rational number p/q with q odd, there is a family Vp/q,Vp/q,Vp,q of curves contained in F^ that are permuted by φ^.

Proof:

We may subdivide A into six rectangular regions as shown in Fig. 3. It is proved in ref. 21 that F^ is a 3-fold branched cover of a 2-sphere S with four branch points. By naturally identifying S with a 4-punctured sphere constructed by gluing two unit squares along their edges, there is a unique isotopy class of curve cp/q with slope p/q in S. Let ρ:FS denote the covering map. Scharlemann proves that ρ1(cp,q)=Vp/q,Vp/q,Vp,q, and these curves are permuted by φ^. □
We note that any 2-component sublink of Vp/qVp/qVp,q is a Casson–Gordon derivative for Q corresponding to some handlebody extension of φ. Fig. 3 shows the three lifts, V3/7, V3/7, and V3/7, of the rational curve 3/7 to the fiber F of the square knot. Observe that φ^6 is the identity map, and φ^3 maps Vp/q to itself but with reversed orientation.
Finally, we can define the GST links. Lemma 16 is also from ref. 21.

Lemma 16.

The GST link Ln is handleslide equivalent to QVn/2n+1. The R-link Ln has property R for n=0,1,2 and is not known to have property R for n3.
For ease of notation, let Vn=Vn/2n+1 and Vn=Vn/2n+1, so that Ln=QVn. Two links L and L are said to be stably handleslide equivalent or just stably equivalent if there are integers n and n so that LUn is handleslide equivalent to LUn. While we can find admissible surfaces for Ln, there is a simpler construction for a family of links Ln stably equivalent to Ln for each n, and we note a link L has stable property R if and only if every link stably equivalent to L has stable property R.

Lemma 17.

The link Ln=QVn is stably equivalent to Ln=VnVn.

Proof:

We show that both links are stably equivalent to QVnVn. Since φ^(Vn)=Vn, we have that Vn is isotopic to Vn in Y^. Carrying this isotopy into S3, we see that after some number of handleslides of Vn over Q, the resulting curve C is isotopic to Vn. Now C can be slid over Vn to produce a split unknot U1, and QVnVn is handleslide equivalent to LnU1. On the other hand, Vn and Vn are homologically independent in the genus two surface F. Thus, there is a sequence of slides of Q over Vn and Vn in F converting Q to a split unknot, so QVnVn is handleslide equivalent to LnU1 as well. □
Next, we define an admissible surface for Ln. Consider a collar neighborhood F×I of F, and let NS3 denote the embedded 3-manifold obtained by crushing F×I to a single curve. Letting Σ=N, we see that Σ is two copies of F, call them F0 and F1, glued along the curve Q.

Lemma 18.

Consider Ln embedded in F0, and push Ln slightly into N. Then Σ is an admissible surface for Ln.

Proof:

First, F×I is a genus four handlebody, as is N, since N is obtained by crushing the vertical boundary of F×I. Moreover, since the exterior E(Q) is fibered with fiber F, we may view this fibering as an open-book decomposition of S3 with binding Q, and thus S3\N¯ is homeomorphic to N, so that Σ is a Heegaard surface for S3.
It remains to be seen that there is a core of N containing Ln, but it suffices to show that there is a pair Dn and Dn of dualizing disks for Ln in N. Note that for any properly embedded arc aF0, there is a compressing disk D(a) for N obtained by crushing the vertical boundary of the disk, a×IF×I. Let a0 and a0 be disjoint arcs embedded in F0 such that a0 meets Vn once and avoids Vn, and a0 meets Vn once and avoids Vn. Then D(a0) and D(a0) are dualizing disks for Ln, completing the Proof. □
Lemma 18 does more than simply prove Σ is admissible; it provides the key ingredients we need to construct a diagram for T(Ln,Σ): Let a1 and a1 denote parallel copies of a0 and a0, respectively, in F1, so that D(a0)=a0a1 and D(a0)=a0a1. By Lemma 4, there is a genus four trisection diagram (α,β,γ) for T(Ln,Σ) so that
β1=D(a0)  β2=D(a0)  γ1=Vn  γ2=Vn.
Noting that (β,γ) defines a genus four splitting of Y2, it follows that any curve disjoint from β1β2γ1γ2 that bounds a disk in either of Hβ or Hγ also bounds in the other handlebody. Let b0 and b0 denote nonisotopic disjoint arcs in F0 that are disjoint from a0a0Ln. Then b0b1 and b0b1 bound disks in N; thus, letting
β3=γ3=b0b1  β4=γ4=b0b1,
we have that (β,γ) is a standard diagram, corresponding to two of the cut systems in a diagram for T(Ln,Σ). To find the curves in α, let N=S3\N¯, and observe that N also has the structure of F×I crushed along its vertical boundary, and N=N=F0F1.
One way to reconstruct S3 from N and N, both of which are homeomorphic to crushed products F×I, is to initially glue F1N to F1N. The result of this initial gluing is again homeomorphic to a crushed product F×I. The second gluing then incorporates the monodromy, so that F0N is glued to F0N via φ. The result of this gluing is that if a1 is an arc in F1N and D(a1) is the corresponding product disk in N, then D(a1)=a1φ(a0), where a0 is a parallel copy of a1 in F0 (using the product structure of N).
Thus, to find curves in α, we can choose any four arcs in F1 cutting the surface into a planar component and construct their product disks. However, if we wish to a find a diagram with relatively little complication with respect to the β and γ curves we have already chosen, it makes sense to choose those four arcs to be a1, a1, b1, and b1. Thus,
α1=a1φ(a0)α3=b1φ(b0)α2=a1φ(a0)α4=b1φ(b0).
We have proved the following.

Proposition 19.

The triple (α,β,γ) forms a (4;0,2,2)-trisection diagram for T(Ln,Σ).
The diagram (α,β,γ) is depicted in Fig. 4. A generalization of this construction allows us to replace Q with any knot of the form T(p,q)#T(p,q).
Fig. 4.
A trisection diagram for T(L3,Σ). Top row shows two copies of F0, along with arcs: φ(a0) and φ(a0) (red), φ(b0) and φ(b0) (pink), b0 and b0 (dark blue), b1 and b1 (light blue), and V3 (dark green) and V3 (light green). Bottom row shows two copies of F1, along with arcs: a1 and a1 (red and dark blue) and b1 and b1 (pink and light blue). The surfaces in the Top row are identified with those in the Bottom row along the oriented puncture. Thus, each column describes the closed genus four surface Σ. Left column encodes a 4-tuple of curves on this surface, namely, α. Right column encodes the 4-tuple β (shades of blue), as well as the two curves γ1 and γ2. The trisection diagram for T(L3,Σ) is obtained by overlaying the two columns. (Note that γ3=β3 and γ4=β4.)

5. A Rectangle Condition for Trisection Diagrams

In this section, we introduce a tool for potential future use. This tool is an adaptation to the setting of trisection diagrams of the rectangle condition for Heegaard diagrams, which was introduced by Casson and Gordon (14) (also ref. 22). A collection of 3g3 pairwise disjoint and nonisotopic curves in a genus g surface Σ is called a pants decomposition, as the curves cut Σ into 2g2 thrice-punctured spheres, or pairs of pants. A pants decomposition defines a handlebody in the same way a cut system does, although a cut system is a minimal collection of curves defining a handlebody, whereas a pants decomposition necessarily contains superfluous curves. An extended Heegaard diagram is a pair of pants decompositions (α+,β+) determining a Heegaard splitting Hα+Hβ+. An extended trisection diagram is a triple of pants decompositions (α+,β+,γ+) determining the spine Hα+Hβ+Hγ+ of a trisection.
Suppose that α+ and β+ are pants decompositions of Σ, and let Pα+ be a component of Σ\ν(α+) and Pβ+ be a component of Σ\ν(β+). Let a1, a2, and a3 denote the boundary components of Pα+ and b1, b2, and b3 denote the boundary components of Pβ+. We say that the pair (Pα+,Pβ+) is saturated if for all i,j,k,l1,2,3, ij, kl, the intersection Pα+Pβ+ contains a rectangle Ri,j,k,l with boundary arcs contained in ai, bk, aj, and bl (Fig. 5, Left). We say that that pair of pants Pα+ is saturated with respect to β+ if for every component Pβ+ of Σ\ν(β+), the pair (Pα+,Pβ+) is saturated.
Fig. 5.
(Left) A pair of pants Pα+ that is saturated with respect to a second pair of pants Pβ+. (Center) A depiction of the contradiction incurred under the assumption that δα+ but δβ+. (Right) A depiction of the contradiction incurred under the assumption that δα+β+γ+.
An extended Heegaard diagram (α+,β+) satisfies the Casson–Gordon rectangle condition if for every component Pα+ of Σ\ν(α+), we have that Pα+ is saturated with respect to β+. Casson and Gordon (14) proved the following.

Theorem 20.

Suppose that an extended Heegaard diagram (α+,β+) satisfies the rectangle condition. Then the induced Heegaard splitting Hα+Hβ+ is irreducible.
Now, let (α+,β+,γ+) be an extended trisection diagram. We say that (α+,β+,γ+) satisfies the rectangle condition if for every component Pα+ of Σ\ν(α+), we have that either Pα+ is saturated with respect to β+ or Pα+ is saturated with respect to γ+.

Remark 21.

Note that since (α+,β+) and (α+,γ+) are extended Heegaard diagrams for the standard manifolds Yk1 and Yk3, it is not possible for either pair to satisfy the rectangle condition of Casson and Gordon (14). In other words, it is not possible that every component Pα+ of α+ be saturated with respect to, say, β+.

Proposition 22.

Suppose that an extended trisection diagram (α+,β+,γ+) satisfies the rectangle condition. Then the induced trisection T with spine Hα+Hβ+Hγ+ is irreducible.

Proof:

Suppose by way of contradiction that T is reducible. Then there exists a curve δΣ=Hα+ that bounds disks D1Hα+, D2Hβ+, and D3Hγ+. Let Dα+ denote the set of 3g3 disks in Hα+ bounded by the curves α+, and define Dβ+ and Dγ+ similarly. There are several cases to consider. First, suppose that δα+, so that D1Dα+, and let Pα+ be a component of Σ\ν(α+) that contains δ as a boundary component. Suppose without loss of generality that Pα+ is saturated with respect to β+. Then, for any curve bβ+, we have that b is the boundary of a component Pβ+ of Σ\ν(β+), where Pα+Pβ+ contains a rectangle with boundary arcs in δ and b. It follows that δ meets every curve bβ+, so δβ+.
Suppose that D2 and Dβ+ have been isotoped to intersect minimally, so that these disks meet in arcs by a standard argument. There must be an outermost arc of intersection in D2, which bounds a subdisk of D2 with an arc δδ, and δ is a wave (an arc with both endpoints on the same boundary curve) contained in a single component Pβ+ of Σ\ν(β+). Let b1 and b2 be the boundary components of Pβ+ disjoint from δ. Since Pα+ is saturated with respect to β+, there is a rectangle RPα+Pβ+ with boundary arcs contained in b1, δ, b2, and some other curve in Pα+ (Fig. 5, Center). Let δ be the arc component of R contained in δ. Since the wave δ separates b1 from b2 in Pβ+, it follows that δδ, a contradiction.
In the second case, suppose that δ is a curve in β+. Note that the Heegaard splitting determined by (α+,γ+) is reducible, and thus by the contrapositive of the Casson–Gordon rectangle condition, there must be some pair of pants Pα+ of Σ\ν(α+) such that Pα+ is not saturated with respect to γ+, so that Pα+ is saturated with respect to β+. Let Pβ+ be a component of Σ\ν(β+) that contains δ as a boundary component. By the above argument, δα+, and if we intersect D1 with Dα+, we can run an argument parallel to the one above to show that δ has a self-intersection, a contradiction. A similar argument shows that δγ+.
Finally, suppose that δ is not contained in any of α+, β+, or γ+. By intersecting the disks D1 and Dα+, we see that there is a wave δδ contained in some pants component Pα+ of Σ\ν(α+). Suppose without loss of generality that Pα+ is saturated with respect to β+. By intersecting D2 with Dβ+, we see that there is a wave δδ contained in some pants component Pβ+ of Σ\ν(β+). Let a1 and a2 be the components of Pα+ that avoid δ, and let b1 and b2 be the components of Pβ+ that avoid δ. By the rectangle condition, Pα+Pβ+ contains a rectangle R whose boundary is made of arcs in a1, b1, a2, and b2. As such, δR contains an arc connecting b1 to b2, while δR contains an arc connecting a1 to a2, but this implies that δδ, a contradiction. We conclude that no such curve δ exists. □
Of course, at this time, the rectangle condition is a tool without an application, which elicits the following question.

Question 23.

Is there an extended trisection diagram that satisfies the rectangle condition?
Note that while it is easy to find three pants decompositions that satisfy the rectangle condition, the difficulty lies in finding three such pants decompositions which also determine a trisection; in pairs, they must be extended Heegaard diagrams for the 3-manifolds Yki.

Acknowledgments

The authors thank Tye Lidman, whose expressed interest in the connections between trisections and Dehn surgery motivated this article. The authors are grateful to Rob Kirby for comments that clarified the exposition of the article and to the anonymous referee for the thorough reading of the manuscript. J.M. is supported by NSF Grants DMS-1400543 and DMS-1758087, and A.Z. is supported by NSF Grant DMS-1664578 and NSF Established Program to Stimulate Competitive Research Grant OIA-1557417.

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Information & Authors

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Published in

Go to Proceedings of the National Academy of Sciences
Go to Proceedings of the National Academy of Sciences
Proceedings of the National Academy of Sciences
Vol. 115 | No. 43
October 23, 2018
PubMed: 30348796

Classifications

Submission history

Published online: October 22, 2018
Published in issue: October 23, 2018

Keywords

  1. Dehn surgery
  2. trisections
  3. 4-manifolds
  4. Generalized Property R Conjecture
  5. Heegaard splitting

Acknowledgments

The authors thank Tye Lidman, whose expressed interest in the connections between trisections and Dehn surgery motivated this article. The authors are grateful to Rob Kirby for comments that clarified the exposition of the article and to the anonymous referee for the thorough reading of the manuscript. J.M. is supported by NSF Grants DMS-1400543 and DMS-1758087, and A.Z. is supported by NSF Grant DMS-1664578 and NSF Established Program to Stimulate Competitive Research Grant OIA-1557417.

Notes

This article is a PNAS Direct Submission.

Authors

Affiliations

Jeffrey Meier1 [email protected]
Department of Mathematics, University of Georgia, Athens, GA 30602;
Alexander Zupan
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588

Notes

1
To whom correspondence should be addressed. Email: [email protected].
Author contributions: J.M. and A.Z. designed research, performed research, and wrote the paper.

Competing Interests

The authors declare no conflict of interest.

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    Characterizing Dehn surgeries on links via trisections
    Proceedings of the National Academy of Sciences
    • Vol. 115
    • No. 43
    • pp. 10817-E10286

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