Characterizing Dehn surgeries on links via trisections
Jeffrey Meier Alexander Zupan
math.GT




empty




arabic










Department of Mathematics, Indiana University, 
Bloomington, IN 47408
jlmeier@indiana.edu
http://pages.iu.edu/ jlmeier 


Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588
zupan@unl.edu
http://www.math.unl.edu/ azupan2


	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 addition, we describe how the potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of the standard four-sphere that are unlikely to be standard.  Finally, we give an analog of the Casson-Gordon Rectangle Condition for trisections that can be used to obstruct reducibility of a given trisection.


Characterizing Dehn surgeries on links via trisections
    Alexander Zupan
    December 30, 2023
======================================================







§ OUTLINE




The purpose of this note is to use both new and existing results to make clear the significant role of the trisection theory of smooth 4-manifolds in the classification of Dehn surgeries on links.  The theory of Dehn surgery on knots has been thoroughly developed over the past forty years.  In general, this research has focused on two major questions:  First, which manifolds can be obtained by a surgery on a knot in a given manifold Y?  Second, given a pair of manifolds Y and Y', for which knots K ⊂ Y 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 years, the Heegaard Floer homology theories of Ozsváth and Szabó have dramatically deepened our collective understanding of Dehn surgeries on knots (see, for instance, <cit.>).

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 S^3 <cit.>. 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 <cit.>.  Thus, surgery classification of links is necessarily considered up to handleslide equivalence, and tools which rely on the topology of a knot exterior S^3 ∖ν(K) are not nearly as useful, since handleslides can significantly alter this topology.

Understanding link surgeries in particular 3-manifolds is intimately connected to smooth 4-manifold topology.  Every smooth 4-manifold X can be described by a handle decomposition, characterized by the attaching link L for the 2-handles, which is contained in the boundary of the union of the 0- and 1-handles.  In other words, X is associated to a framed link L ⊂#^k(S^1  S^2) such that Dehn surgery on L yields #^k' (S^1  S^2).  Conversely, such a link L (which we will call admissible) describes a handle decomposition of a smooth 4-manifold, which we denote X_L.  Thus, classifying all such surgeries would be equivalent to classifying all smooth 4-manifolds.

Clearly, this is an insurmountable task, but to make the problem more tractable, we consider various restrictions placed on the parameters k, k', and n, where n represents the number of components of the link L.  For example, let k = k' = 0.  In the case that n=1, Gordon and Luecke proved that knots are determined by their complements, and thus the only knot in S^3 that admits an integral S^3 surgery (a cosmetic surgery) is a ± 1-framed unknot <cit.>. In this paper, we will describe the proof of the following theorem from <cit.>.


If L ⊂ S^3 is a two-component link with tunnel number one with an integral surgery to S^3, then L is handleslide equivalent to a 0-framed Hopf link or ± 1-framed unlink.


Another significant case occurs when k = 0 and k' = n.  In other words, we wish to understand n-component links in S^3 with surgeries to #^n (S^1  S^2).  We call such a link L an R-link, noting that R-links correspond precisely to the collection of geometrically simply-connected homotopy 4-spheres, i.e. homotopy 4-spheres built without 1-handles.  The Generalized Property R Conjecture (GPRC), Kirby Problem 1.82 <cit.>, contends that every R-link is handleslide equivalent to a 0-framed unlink.  The conjecture is known to be true in the case n=1 via Gabai's proof of Property R <cit.>.  In <cit.>, the authors, in collaboration with Trent Schirmer, proved a stable version of the GPRC for a class of links.


If L ⊂ S^3 is an n-component R-link with tunnel number n, then the disjoint union of L with a 0-framed unknot is handleslide equivalent to a 0-framed unlink.


As foreshadowed above, the proofs of these theorems are 4-dimensional in nature, utilizing a prominent new tool: trisections of smooth 4-manifolds.  A trisection is a decomposition of a 4-manifold X into three simple pieces, a 4-dimensional version of a 3-dimensional Heegaard splitting.  Elegantly connecting the two theories, Gay and Kirby proved that every smooth 4-manifold admits a trisection, and every pair of trisections for a given 4-manifold have a common stabilization <cit.>, mirroring the Reidemeister-Singer Theorem <cit.> 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 <cit.>.  A trisection is said to be standard if it is an unbalanced stabilization of the genus zero trisection of S^4, and thus, every trisection of S^4 becomes standard after some number of Gay-Kirby stabilizations.
 
In Section <ref>, we describe a process by which an R-link L paired with an admissible Heegaard surface Σ for its exterior is converted to a trisection (L,Σ) of the 4-manifold X_L.  The new main result of this paper is a technical theorem that connects R-links to properties of these trisections.  The terms {2}–standard and {2,3}–standard refer to trisections that become standard after allowing restricted types of unbalanced stabilizations; we will postpone the rigorous definitions for now.


Suppose L is an R-link and Σ is any admissible surface for L.


  * If L satisfies the GPRC, then (L,Σ) is {2}–standard.

  * The link L satisfies the Stable GPRC if and only if (L,Σ) is {2,3}–standard.



In Section <ref>, we analyze examples of Gompf-Schlarlemann-Thompson, the most prominent possible counterexamples to the GPRC.  The first step in a program to disprove the GPRC via Theorem <ref> is to find low-genus admissible surfaces for these links, along with diagrams for their induced trisections.  We outline this process; extensions of Section <ref> will appear in forthcoming work <cit.>.

In Section <ref>, we introduce an analog of the Casson-Gordon Rectangle Condition <cit.> for trisection diagrams, giving a sufficient condition for a trisection diagram to correspond to an irreducible trisection.



 §.§ Acknowledgements
 
The first author is supported by NSF grants DMS-1400543 and DMS-1664540, and the second author is supported by NSF grant DMS-1664578 and NSF-EPSCoR grant OIA-1557417.










§ TRISECTIONS AND ADMISSIBLE LINKS




All manifolds are connected and orientable, unless otherwise stated.  We will let ν( · ) refer to an open regular neighborhood in an ambient manifold that should be clear from context.  The tunnel number of a link L ⊂ Y is the cardinality of the smallest collection of arcs a with the property that Y ∖ν(L ∪ a) is a handlebody.  In this case, ν(L ∪ a) 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 L_1 and L_2.  The framings of L_1, L_2 and a induce an embedded surface S ⊂ Y, homeomorphic to a pair of pants, such that L_1 ∪ L_2 ∪ a is a core of S.  Note that S has three boundary components, two of which are isotopic to L_1 and L_2.  Let L_3 denote the third boundary component, with framing induced by S.  If L' is the framed link (L ∖ L_1) ∪ L_3, we say that L' is obtained from L by a handleslide of L_1 over L_2 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 <cit.>.  Recall that an R-link is an n-component link in S^3 with a Dehn surgery to the manifold #^n(S^1  S^2), which we henceforth denote by Y_n.  Let U_n denote the n-component zero-framed unlink in S^3.  If an R-link L is handleslide equivalent to U_n, we say that L has Property R.  If the split union L ⊔ U_r is handleslide equivalent to U_m for some integers r and m, we say that L has Stable Property R.  The following conjectures are well-known; the first is Kirby Problem 1.82 <cit.>.


Every R-link has Property R.



Every R-link has Stable Property R.


In this section, we explore the relationship between R-links (and a more general family we call admissible 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;k_1,k_2,k_3)–trisection  of X is a decomposition X = X_1∪ X_2∪ X_3 such that
	
		
  * Each X_i is a four-dimensional 1–handlebody, ♮^k_i(S^1× B^3);
		
  * If i≠j, then H_ij = X_i∩ X_j is a three-dimensional handlebody, ♮^g(S^1× D^2); and
		
  * The common intersection Σ = X_1∩ X_2∩ X_3 is a closed genus g surface.
	
	The surface Σ is called the trisection surface, and the parameter g is called the genus of the trisection.    The trisection  is called balanced if k_1=k_2=k_3=k, in which case it is called a (g;k)–trisection; otherwise, it is called unbalanced.  We call the union H_12∪ H_23∪ H_31 the spine of the trisection.  In addition, we observe that ∂ X_i = Y_k_i = H_ij∪_Σ H_li is a genus g Heegaard splitting.  Because there is a unique way to cap off Y_k_i with ♮^k_i(S^1× B^3) <cit.>,  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 a 2g-punctured sphere.  A cut system δ is said to define a handlebody H_δ if each curve in δ bounds a disk in H_δ.  A triple (α,β,γ) of cut systems is called a (g;k_1,k_2,k_3)–trisection diagram for  if α, β, and γ define the components H_α, H_β, and H_γ of the spine of .  We set the convention that H_α = X_3∩ X_1, H_β = X_1∩ X_2, and H_γ = X_2∩ X_3.  The careful reader will note that this convention differs slightly from <cit.>.  With these conventions, (α,β), (β,γ), and (γ,α) are Heegaard diagrams for Y_k_1, Y_k_2, and Y_k_3, respectively.  In <cit.>, Gay and Kirby prove that every smooth 4-manifold admits a trisection, and trisection diagrams, modulo handle slides within the three collections of curves, are in one-to-one correspondence with trisections.


	Trisections with genus at most two are well-understood. See Figure <ref>.
	
		
  * There is a unique genus zero trisection; the (0,0)–trisection describing S^4.
		
  * There are exactly six genus one trisections.  Both ^2 and ^2 admit (1;0)–trisections; S^1× S^3 admits a (1;1)–trisection; and S^4 admits three unbalanced genus one trisections.
		
  * There is a unique irreducible (defined below) genus two trisection <cit.>, which describes S^2× S^2.
	


Given trisections  and ' 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  and ' along the boundary of this neighborhood.  The resulting trisection is uniquely determined in this manner; we denote it by #'.  A trisection  is called reducible if = '#”, where neither ' nor ” is the genus zero trisection; otherwise, it is called irreducible.  Equivalently,  is reducible precisely when there exists a curve δ in Σ that bounds compressing disks in H_, H_, and H_.  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 S^3, and a similar structure exists for trisections.  Let _i denote the unique genus one trisection of S^4 satisfying k_i=1.  Diagrams for these three trisections are shown in Figure <ref>.  A trisection  is called i–stabilized if = '#_i, and is simply called stabilized if it is i–stabilized for some i=1,2,3.  Two trisections ' and ” are called stably equivalent if there is a trisection  that is a stabilization of both ' and ”.  Gay and Kirby proved that any two trisections of a fixed 4-manifold are stably equivalent <cit.>.

We say that a trisection  is standard if  can be expressed as the connected sum of the trisections listed in Examples <ref>.  Theorems in <cit.> classify trisections of genus two.


	Every trisection  with genus g =2 is standard.












Below, we see how this theorem implies Theorem <ref>, and for this purpose, we turn our attention to surgery on links.  




 §.§ Admissible links and surfaces
 


Recall that Y_k denotes #^k(S^1 × S^2), and let L be a framed n–component link in Y_k such that Dehn surgery on L yields Y_k'.  We call such a link admissible.  If L is an admissible link, L describes a closed 4-manifold X_L with a handle decomposition with k 1–handles, n 2–handles, and k' 3–handles.  An admissible Heegaard surface Σ for L is a Heegaard surface cutting Y_k into two handlebodies H and H', such that a core of H contains L.  As such, C = H ∖ν(L) is a compression body and Σ may be viewed as a Heegaard surface for the link exterior E(L) = Y_k ∖ν(L).  Let H_L be the handlebody that results from Dehn filling C (or performing Dehn surgery on L in H) along the framing of the link L.  An admissible pair consists of an admissible link together with an admissible Heegaard surface.

For completeness, we will also allow the empty link, L =.  An admissible surface Σ for the empty link is a (standard) genus g Heegaard surface for Y_k.  A genus g Heegaard diagram (,) for Y_k is called standard if ∩ contains k curves, and the remaining g-k 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.


	Let L be an admissible n-component link in Y_k.  Every admissible pair (L,Σ) gives rise to a trisection (L,Σ) with spine H' ∪ H ∪ H_L.  If g(Σ) = g, then (L,Σ) is a (g;k,g-n,k')-trisection.  Moreover, there is a trisection diagram (,,) for (L,Σ) such that


  * H_ = H', H_ = H, and H_ = H_L;

  * L is a sublink of , where  is viewed as a link framed by Σ in Y_k = H_∪ H_; and

  * (,) is a standard diagram for Y_g-n, where ∩ = ∖ L.




This is proved (in slightly different formats) for L ≠ in both <cit.> and <cit.>.  If L =, then it follows easily that X_L has a handle decomposition without 2-handles, H = H_L, and H' ∪ H ∪ H_L is the spine for the (g;k,g,k)-trisection (L,Σ) of X_L.  In this case, there is a diagram such that =, the standard genus g diagram of Y_g.




This machinery is enough to prove Theorem <ref>, classifying cosmetic surgeries on tunnel number one links in S^3.  Note that the conventions H_ = H', H_ = H, and H_ = H_L agree with our earlier conventions identifying the union of the 0–handle and the 1–handles with X_1, the trace of the Dehn surgery on H_ along L with X_2, and the union of the 3–handles and the 4–handle with X_3.


Suppose L ⊂ S^3 is a tunnel number one link with an integral Dehn surgery to S^3.  Then there exists an admissible surface Σ⊂ S^3 and a genus two trisection (L,Σ) with a diagram (,,), where H_L = H_.  By Lemma <ref>, (L,Σ) is a (2,0)-trisection, the two curves in  are isotopic to the link L in S^3 = H_∪ H_, and the surface framing of  in Σ is the framing of L.  By Theorem <ref>, the trisection (L,Σ) of X_L is standard, and (,,) is handleslide equivalent to a standard diagram (',',').  Since (L,Σ) is (2,0)-trisection, X_L is diffeomorphic to either S^2  S^2 or ±#±.  In the first case,  is handleslide equivalent to ', which is a zero-framed Hopf link in S^3 = H_∪ H_.  In the second case,  is handleslide equivalent to ', a 2-component unlink with framings ± 1 and ± 1, completing the proof.


We now turn our attention to R-links.  Note that if L is an R-link, then the smooth 4-manifold X_L has a handle decomposition with no 1-handles, n 2-handles, and n 3-handles; thus X_L is a simply connected 4-manifold with χ(X_L) = 2, so that X_L is a homotopy S^4.  We describe an immediate connection between Stable Property R and trisections in the next lemma.


Suppose L is an R-link with admissible surface Σ and (L,Σ) is a standard trisection of S^4.  Then L has Stable Property R.



By Lemma <ref>, the trisection (L,Σ) has a diagram (,,) such that (,) is the standard Heegaard diagram for Y_g-n.  Viewing  as a g-component link in S^3 = H_∪ H_, we have that (g-n) curves in  bounds disks in H_, while the remaining n curves are isotopic to L (and are disjoint from the (g-n) disks).  Thus, as a link in S^3, we have = L ⊔ U_g-n.

In addition, the trisection (L,Σ) is a standard (g;0,g-n,n)-trisection of S^4 by hypothesis.  As such, it must be a connected sum of g-n copies of _2 and n copies of _3, and it has a standard diagram, (',','), where g-n curves in ' are also curves in ', and the remaining n curves are also curves in '.  Thus, in S^3 = H_'∪ H_', the curves ' comprise a g-component unlink, with surface framing equal to the zero framing on each component.  Since (,,) and (',',') are trisection diagrams for the same trisection, we have that  is handleslide equivalent to ' via slides contained in Σ.  Thus,  and ' are handleslide equivalent links in S^3.  We conclude that L has Stable Property R, as desired.



Theorem <ref> can be quickly proved using this lemma and the following result from <cit.> as its main input.


Every (g;0,1,g-1)-trisection is a standard trisection of S^4.






Suppose L ⊂ S^3 is an n-component link with tunnel n with a Dehn surgery to #^n(S^1  S^2).  Then by Lemma <ref> there exists an admissible surface Σ⊂ S^3 and an (n+1;0,1,n)-trisection (L,Σ).  By Theorem <ref>, the trisection (L,Σ) is standard, and by Lemma <ref>, L has Stable Property R.  In fact, the proof of Lemma <ref> reveals that L ⊔ U_1 has Property R, as desired.






§ R-LINKS AND STABILIZATIONS




In order to prove the third main theorem, we will further develop the connection between R-links, their induced trisections, and the various stabilization operations.


If L = in Y_k and g(Σ) = g, then X_L = #^k(S^1 × S^3), and (,Σ) is the connected sum of k copies of the standard (1;1)-trisection of S^1 × S^3 and g-k copies of _2.


By Waldhausen's Theorem <cit.>, Y_k has a standard Heegaard diagram, (,), and by Lemma <ref>, (,,) is a trisection diagram for (,Σ).  The k curves in ∩ give rise to k summands of the standard genus one splitting of S^1 × S^3, and the remaining g-k pairs give rise to g-k copies of _2.











In order to understand operations on an admissible link L and Heegaard surface Σ which will correspond to various stabilizations of (L,Σ), we introduce several additional definitions.  Let (L_1,Σ_1) ⊂ Y_k_1 and (L_2,Σ_2) ⊂ Y_k_2 be any admissible pairs, and define the operation ∗ by

    (L_1,Σ_1) ∗ (L_2,Σ_2) = (L_1 ⊔ L_2, Σ_1 #Σ_2),

where the connected sum is taken so that L_1 ⊔ L_2 is not separated by the surface Σ_1 #Σ_2.  Note that (L_1,Σ_1) ∗ (L_2,Σ_2) ⊂ Y_k_1+k_2.


If (L_1,Σ_1) and (L_2,Σ_2) are admissible pairs, then (L,Σ) = (L_1,Σ_1) ∗ (L_2,Σ_2) is an admissible pair, and (L,Σ) = (L_1,Σ_1) #(L_2,Σ_2).


It is clear that the framed link L_1 ⊔ L_2 ⊂ Y_k_1+k_2 has the appropriate surgery.  Suppose Σ_i bounds a handlebody H_i with core C_i containing L_i.  Then there is a core C for H_1 ♮ H_2 such that L_1 ⊔ L_2 ⊂ C_1 ⊔ C_2 ⊂ C, and thus Σ_1 #Σ_2 is admissible as well.  For the second claim, note that the curve δ arising from the connected sum Σ = Σ_1 #Σ_2 is a reducing curve for (L,Σ), splitting it into the trisections (L_1,Σ_1) and (L_2,Σ_2).


Let U be a 0-framed unknot in S^3, and let Σ_U be the genus one splitting of S^3 such that one of the solid tori bounded by Σ_U contains U as a core.  In addition, let W denote the knot S^1 ×{pt}⊂ S^1 × S^2 with framing given by the fibering, and let Σ_W be the genus one splitting of S^1 × S^2 such that one of the solid tori bounded by Σ_W contains W as a core.  Note that both (U,Σ_U) and (W,Σ_W) are admissible pairs.  Finally, let Σ_ be the genus one Heegaard surface for S^3, to be paired with the empty link.


The links (W,Σ_W), (,Σ_), and (U,Σ_U) yield the following trisections:


  * (W,Σ_W) = _1.

  * (,Σ_) = _2.

  * (U,Σ_U) = _3.



First, note that each trisection in question has genus one.  Since framed surgery on W ⊂ Y_1 yields S^3, by Lemma <ref>, (W,Σ_W) is a (1;1,0,0)–trisection and must be _1. Similarly, (,Σ_) is a (1;0,1,0)–trisection and must be _2.  Finally, framed surgery on U yields Y_1, so (U,Σ_U) is a (1;0,0,1)–trisection and must be _3.


By combining Lemmas <ref>, <ref>, and <ref>, we obtain

Suppose (L,Σ) is an admissible link, with = (L,Σ).


  * ((L,Σ) ∗ (W,Σ_W)) is the 1–stabilization of .

  * ((L,Σ) ∗ (,Σ_)) is the 2–stabilization of .

  * ((L,Σ) ∗ (U,Σ_U)) is the 3–stabilization of .

In addition, if Σ_+ is the stabilization of Σ (as a Heegaard surface for Y_k), then (L,Σ_+) = (L,Σ) ∗ (,Σ_).


We say that two trisections _1 and _2 of a 4-manifold X are 2–equivalent if there is a trisection  that is the result of 2–stabilizations performed on both _1 and _2.


	If Σ_1 and Σ_2 are two distinct admissible surfaces for an admissible link L, then the trisections (L,Σ_1) and (L,Σ_2) are 2–equivalent.



	Since both Σ_1 and Σ_2 are Heegaard surfaces for E(L), they have a common stabilization Σ by the Reidemeister-Singer Theorem <cit.>.  By Lemma <ref>, the surface Σ is admissible, and by Corollary <ref>, (L,Σ) can be obtained by 2–stabilizations of (L,Σ_i).


Since 2–equivalence is an equivalence relation, Lemma <ref> implies that every admissible surface Σ for an admissible link L belongs to the same 2–equivalence class.  Hence, L has a well-defined 2–equivalence class; namely, the 2–equivalence class of (L,Σ).  If two admissible links L_1 and L_2 give rise to 2–equivalent trisections, we say that L_1 and L_2 are 2–equivalent.

Suppose that L is an n-component admissible link with admissible surface Σ, so that Σ cuts Y_k into H ∪ H', and L is isotopic into a core C ⊂ H.  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 (,,) is the trisection diagram for (L,Σ) guaranteed by Lemma <ref>, then the n disks bounded by the n curves in  that are not in  are a set of dualizing disks for L.


	If admissible links L_1 and L_2 are related by a handleslide, then L_1 and L_2 are 2–equivalent.


	If L_i is an n–component link, then L_1 and L_2 have n-1 components in common and differ by a single component, L_1' ⊂ L_1 and L_2' ⊂ L_2, where a slide of L_1' over another component L' of L_1 along a framed arc a yields L_2'.  Consider Γ = L_1 ∪ a, an embedded graph with n-1 components, and let Σ be a Heegaard surface cutting S^3 into H ∪ H', where Γ is contained in a core of H.  Then L_1 is also contained in a core of H, and Σ is admissible (with respect to L_1).  Let {D_1} be a set of dualizing disks for L_1.  A priori, the arc a might meet some of the disks in {D_1}; however, if this is the case, we can perform a sequence of stabilizations on Σ, after which a avoids all of the disks {D_1}.  Thus, we suppose without loss of generality that a ∩{D_1} = ∅.
	
There is an isotopy taking Γ into Σ, preserving the intersections of L_i with the dualizing disks {D_1}, so that the framing of Γ agrees with its surface framing in Σ.  As such, we can perform the handleslide of L_1' over L' along a within the surface Σ, so that the resulting link L_2 is also contained in Σ, with framing given by the surface framing.  Let D_1' ∈{D_1} be the disk that meets L_1' once, and let D' ∈{D_1} be the disk that meets L' once.  There is an arc a', isotopic in Σ to an arc in Γ, that connects D_1' to D'. See Figure <ref>. Let D_2' be the compressing disk obtained by banding D_1' to D' along a'.  Then {D_2} = ({D_1}∖ D')∪ D_2' is a set of dualizing disks for L_2. Thus, by pushing L_2 back into H, we see that Σ is an admissible surface for L_2.



Following Lemma <ref>, let H_i ∪ H_i' ∪ H_L_i be a spine for (L_i,Σ).  By construction, H_1 = H_2 and H_1' = H_2'.  Finally, since H_i is Dehn surgery on L_i in H_i , and L_1 and L_2 are related by a single handleslide, we have H_L_1 = H_L_2.  It follows that (L_1,Σ) = (L_2,Σ), and we conclude that L_1 and L_2 are 2–equivalent.


For the rest of the section, we will restrict our attention to admissible links in S^3.  Let U_n denote the zero-framed, n–component unlink, so X_U_n = S^4.  Recall that a standard trisection of S^4 is the connected sum of copies of _1, _2, and _3.


	Let Σ be any admissible surface for U_n, then (U_n,Σ) is standard.


We induct on (n,g) with the dictionary ordering.  If n=1, then E(U_1) is a solid torus.  If g=1, then Σ = Σ_U, so that (U_1,Σ_U) = _3 by Lemma <ref>.  If n=1 and g > 1, then Σ is stabilized <cit.> (see also <cit.>), which means that (U_1,Σ) is 2–stabilized by Corollary <ref>, and as such, (U_1,Σ) is standard by induction.

In general, note that the Heegaard genus of an n–component unlink is n; thus g ≥ n for all possible pairs (n,g).  For n>1, we have that E(U) is reducible, and so Haken's Lemma <cit.> implies that Σ is reducible, splitting into the connected sum of genus g_1 and g_2 surfaces Σ_1 and Σ_2, where Σ_i is a Heegaard surface for E(U_n_i).  Then (U_n,Σ) = (U_n_1,Σ_1) #(U_n_2,Σ_2), where (n_i,g_i) < (n,g).  Since both summands are standard trisections by induction, it follows that (U_n,Σ) is also standard, completing the proof.




A trisection  is said to be 2–standard if it becomes standard after some number of 2-stabilizations.  Similarly,  is {2,3}–standard if it becomes standard after some number of 2- and 3-stabilizations.


Suppose L has Property R. By Lemma <ref>, L and U_n are 2–equivalent links.  Thus, (L,Σ) is 2–equivalent to some trisection coming from U_n, but all trisections induced by U_n are standard by Lemma <ref>, and thus (L,Σ) becomes standard after a finite sequence of 2–stabilizations.
	
If L has Stable Property R, then L ⊔ U_n has Property R for some n, and thus ((L,Σ) ∗ (U,Σ_U) ∗…∗ (U,Σ_U)) is 2–standard by the above arguments.  By Lemma <ref> and Corollary <ref>,

    ((L,Σ) ∗ (U,Σ_U) ∗…∗ (U,Σ_U)) = (L,Σ) #_3 #…#_3;

hence (L,Σ) is {2,3}–standard.

Finally, if the trisection (L,Σ) is {2,3}–standard, then there exist integers s and t such that the connected sum of (L,Σ) with s copies of _2 and t copies of _3 is standard.  Let (L_*,Σ_*) be the admissible pair given by

    (L_*,Σ_*) = (L,Σ) ∗(,Σ_) ∗…∗ (,Σ_)_s  ∗(U,Σ_U) ∗…∗ (U,Σ_U)_t.

By assumption, (L_*,Σ_*) is standard, so by Lemma <ref>, the link L_* has Stable Property R.  But by definition of ∗, we have L_* = L ⊔ U_t, and thus L also has Stable Property R, completing the proof.


























§ TRISECTING THE GOMPF-SCHARLEMANN-THOMPSON EXAMPLES



Although this view has changed in the past and may change in the future, it is the current view of the authors that the GPRC is likely false.  In light of this opinion, we will outline the first steps one might take to employ Theorem <ref> to disprove the GRPC or the Stable GPRC.  Let L be an R-link with admissible surface Σ.  By Theorem <ref>, if (L,Σ) is not {2}–standard, then L fails to have Property R.  Thus, we in this section we will show how to take the most promising potential counterexamples to the GPRC and construct admissible surfaces and their corresponding trisections.

The possible counterexamples mentioned in the previous paragraph were produced by Gompf-Scharlemann-Thompson <cit.>, building on work of Akbulut-Kirby <cit.>.   We will call this family the GST links.  In order to describe the construction of the GST links, we need several preliminary details.  Let Q denote the square knot, the connected sum of the right-handed and left-handed trefoil knots, and let F denote the genus two fiber surface for the square knot.  In <cit.>, Scharlemann depicted an elegant way to think about the monodromy corresponding to the fibration of E(Q) by F:  We may draw F as a topological annulus A, and such that


  * A disk D has been removed from A,

  * each component of A is split into six edges and six vertices, and

  * opposite inside edges and opposite outside edges of A are identified to form 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.  Let Y_Q be the closed 3-manifold obtained by 0-surgery on Q, so that Y_Q is a fibered 3-manifold with fiber surface F and monodromy φ, called the closed monodromy of Q. Note that the monodromy F is an honest 1/6th rotation of the annulus in Figure <ref>, since, in this case, the puncture has been filled in by the Dehn surgery.  Details can be found in <cit.> and <cit.>, where the following lemma is first proved.


For every rational number p/q with q odd, there is a family {V_p/q,V'_p/q,V”_p,q} of curves contained in F that are permuted by φ.



We may subdivide A into six rectangular regions as shown in Figure <ref>.  It is proved in <cit.> that F is a 3-fold branched cover of a 2-sphere S with four branch points.  By naturally identifying S with 4-punctured sphere constructed by gluing two unit squares along their edges, there is a unique isotopy class of curve c_p/q with slope p/q in S.  Let ρ:F → S denote the covering map.  Scharlemann proves that ρ^-1(c_p,q) = {V_p/q,V'_p/q,V”_p,q}, and these curves are permuted by φ.


Figure <ref> shows the three lifts, V_3/7, V_3/7', and V_3/7”, of the rational curve 3/7 to the fiber F of the square knot.  Note that φ  ^6 is the identity map, and φ ^3 maps V_p/q to itself but with reversed orientation.



Finally, we can define the GST links.  The next lemma is also from <cit.>.


The GST link L_n is handleslide equivalent to Q ∪ V_n/2n+1.  The R-link L_n has Property R for n = 0,1,2 and is not known to have Property R for n ≥ 3.


For ease of notation, let V_n = V_n/2n+1 and V_n' = V'_n/2n+1, so that L_n = Q ∪ V_n.  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 L ⊔ U_n is handleslide equivalent to L' ⊔ U_n'.  While we can find admissible surfaces for L_n, there is a simpler construction for a family of links L_n' stably equivalent to L_n 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.


The link L_n = Q ∪ V_n is stably equivalent to L'_n = V_n ∪ V_n'.


We will show that both links are stably equivalent to Q ∪ V_n ∪ V_n'.  Since φ(V_n) = V_n', we have that V'_n is isotopic to V_n in Y_Q.  Carrying this isotopy into S^3, we see that after some number of handleslides of V_n' over Q, the resulting curve C' is isotopic to V_n.  Now C' can be slide over V_n to produce a split unknot U_1, and Q ∪ V_n ∪ V_n' is handleslide equivalent to L_n ⊔ U_1.  On the other hand, V_n and V_n' are homologically independent in the genus two surface F.  Thus, there is a sequence of slides of Q over V_n and V_n' converting it to a split unknot, so Q ∪ V_n ∪ V_n' is handleslide equivalent to L_n' ⊔ U_1 as well.


Next, we will define an admissible surface for L_n'.  Consider a collar neighborhood F  I of F, and let N ⊂ S^3 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 F_0 and F_1, glued along the curve Q.


Consider L_n' embedded in F_0, and push L_n' slightly into N.  Then Σ is an admissible surface for L_n'.


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 S^3 with binding Q, and thus S^3 ∖ N is homeomorphic to N, so that Σ is a Heegaard surface for S^3.

It remains to be seen that there is a core of N containing L_n', but it suffices to show that there is a pair D_n and D_n' of dualizing disks for L_n' in N.  Note that for any properly embedded arc a ⊂ F_0, there is a compressing disk D(a) for N obtained by crushing the vertical boundary of the disk a  I ⊂ F  I.  Let a_0 and a_0' be disjoint arcs embedded in F_0 such that a_0 meets V_n once and avoids V_n', and a_0' meets V_n' once and avoids V_n.  Then D(a_0) and D(a_0') are dualizing disks for L_n', completing the proof.


Lemma <ref> does more than simply prove Σ is admissible; it provides the key ingredients we need to construct a diagram for (L_n',Σ):  Let a_1 and a_1' denote parallel copies of a_0 and a_0', respectively, in F_1, so that D(a_0) = a_0 ∪ a_1 and D(a_0') = a_0' ∪ a_1'.  By Lemma <ref>, there is a genus four trisection diagram (,,) for (L_n',Σ) so that

    _1 =  D(a_0)     _2 =  D(a_0')     _1 = V_n     _2 = V_n'.

Noting that (,) defines a genus four splitting of Y_2, it follows that any curve disjoint from _1 ∪_2 ∪_1 ∪_2 that bounds a disk in either of H_ or H_ also bounds in the other handlebody.  Let b_0 and b_0' denote non-isotopic disjoint arcs in F_0 that are disjoint from a_0 ∪ a_0' ∪ L_n'.  Then b_0 ∪ b_1 and b_0' ∪ b_1' bound disks in N; thus letting

    _3 = _3 = b_0 ∪ b_1     _4 =  _4 = b_0' ∪ b_1',

we have that (,) is a standard diagram, corresponding to two of the cut systems in a diagram for (L_n,Σ).  To find the curves in , let N' = S^3 ∖ N, and observe that N' also has the structure of F  I crushed along its vertical boundary, and N' =  N = F_0 ∪ F_1.



One way to reconstruct S^3 from N and N', both of which are homeomorphic to crushed products F  I, is to initially glue F_1 ⊂ N' to F_1 ⊂ N'.  The result of this initial gluing is again homeomorphic to a crushed product F  I.  The second gluing then incorporates the monodromy, so that F_0 ⊂ N' is glued to F_0 ⊂ N via φ.  The result of this gluing is that if a_1 is an arc in F_1 ⊂ N' and D'(a_1) is the corresponding product disk in N', then D'(a) = a_1 ∪φ(a_0), where a_0 is a parallel copy of a_1 in F_0 (using the product structure of N).

Thus, in order to find curves in , we can choose any four arcs in F_1 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  and  curves we have already chosen, it makes sense to choose those four arcs to be a_1, a_1', b_1, and b_1'.  Thus,

    _1 = a_1 ∪φ(a_0)     _2 = a_1' ∪φ(a_0')     _3 = b_1 ∪φ(b_0)     _4 = b_1' ∪φ(b_0').


We have proved the following:


The triple (,,) forms a (4;0,2,2)-trisection diagram for (L_n,Σ).








§ A RECTANGLE CONDITION FOR TRISECTION DIAGRAMS




In the final section, we introduce a tool for potential future use.  This tool is an adaptation of the Rectangle Condition for Heegaard diagrams of Casson and Gordon <cit.> to the setting of trisection diagrams.  A collection of 3g-3 pairwise disjoint and nonisotopic curves in a genus g surface Σ is called a pants decomposition, as the curves cut Σ into 2g-2 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 (,) determining a Heegaard splitting H_∪ H_.  An extended trisection diagram is a triple of pants decompositions (,,) determining the spine H_∪ H_∪ H_ of a trisection.

Suppose that  and  are pants decompositions of Σ, and let P_ be a component of Σ∖ν() and P_ a component of Σ∖ν().  Let a_1, a_2, and a_3 denote the boundary components of P_; b_1, b_2, and b_3 the boundary components of P_.  We say that the pair (P_,P_) is saturated if for all i,j,k,l ∈{1,2,3}, i≠ j, k ≠ l, the intersection P_∩ P_ contains a rectangle R_i,j,k,l with boundary arcs contained in a_i, b_k, a_j, and b_l.  We say that that pair of pants P_ is saturated with respect to  if for every component P_ of Σ∖ν(), the pair (P_,P_) is saturated.

An extended Heegaard diagram (,) satisfies the Rectangle Condition of Casson-Gordon if for every component P_ of Σ∖ν(), we have P_ is saturated with respect to .  Casson and Gordon proved the following.


Suppose that an extended Heegaard diagram (,) satisfies the Rectangle Condition.  Then the induced Heegaard splitting H_∪ H_ is irreducible.


Now, let (,,) be an extended trisection diagram.  We say that (,,) satisfies the Rectangle Condition if for every component P_ of Σ∖ν(), we have that either P_ is saturated with respect to  or P_ is saturated with respect to .


Suppose that an extended trisection diagram satisfies the Rectangle Condition.  Then the induced trisection  with spine H_∪ H_∪ H_ is irreducible.


Suppose by way of contradiction that  is reducible.  Then there exists a curve δ⊂Σ =  H_ that bounds disks D_1 ⊂ H_, D_2 ⊂ H_, and D_3 ⊂ H_.  Let D_ denote the set of 3g-3 disks in H_ bounded by the curves , and define D_ and D_ similarly.  There are several cases to consider.  First, suppose that δ∈, so that D_1 ∈ D_, and let P_ be a component of Σ∖ν() that contains δ as a boundary component.  Suppose without loss of generality that P_ is saturated with respect to .  Then, for any curve b ∈, we have that b is the boundary of a component P_ of Σ∖ν(), where P_∩ P_ contains a rectangle with boundary arcs in δ and b.  It follows that δ meets every curve b ∈, so δ∉.

Suppose that D_2 and D_ 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 D_2, which bounds a subdisk of D_2 with an arc δ' ⊂δ, and δ' is a wave (an arc with both endpoints on the same boundary curve) contained a single component P_ of Σ∖ν().  Let b_1 and b_2 be the boundary components of P_ disjoint from δ'.  Since P_ is saturated with respect to , there is rectangle R ⊂ P_∩ P_ with boundary arcs contained in b_1, δ, b_2, and some other curve in P_.  Let δ” be the arc component of R contained in δ.  Since the wave δ' separates b_1 from b_2 in P_, it follows that δ' ∩δ”≠, a contradiction.

In the second case, suppose that δ is a curve in .  Note that the Heegaard splitting determined by (,) is reducible, and thus by the contrapositive of Casson-Gordon's rectangle condition, there must be some pants decomposition P_ of Σ∖ν() such that P_ is not saturated with respect to , so that P_ is saturated with respect to β.  Let P_ be a component of Σ∖ν() that contains δ as boundary component.  By the above argument, δ∉, and if we intersect D_1 with D_ we can run an argument parallel to the one above to show that δ has a self-intersection, a contradiction.  A parallel argument shows that δ∉.

Finally, suppose that δ is not contained in any of , , or .  By intersecting the disks D_1 and D_, we see that there is a wave δ' ⊂δ contained in some pants component P_ of Σ∖ν().  Suppose without loss of generality that P_ is saturated with respect to .  By intersecting D_2 with D_, we see that there is a wave δ”⊂δ contained in some pants component P_ of Σ∖ν().  Let a_1 and a_2 be the components of P_ that avoid δ', and let b_1 and b_2 be the components of P_ that avoid δ”.  By the Rectangle Condition, P_∩ P_ contains a rectangle R whose boundary is made of arcs in a_1, b_1, a_2, and b_2.   As such, δ' ∩ R contains an arc connecting b_1 to b_2, while δ”∩ R contains an arc connecting a_1 to a_2, but this implies that δ' ∩δ”≠, 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:


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 Y_k.



amsalpha








