Closed essential surfaces and weakly reducible Heegaard splittings in ...

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Journal of Knot Theory and Its Ramifications Vol. 13, No. 6 (2004) 829–843 c World Scientific Publishing Company

CLOSED ESSENTIAL SURFACES AND WEAKLY REDUCIBLE HEEGAARD SPLITTINGS IN MANIFOLDS WITH BOUNDARY

YOAV MORIAH Department of Mathematics, Technion, Haifa 32000, Israel [email protected] ERIC SEDGWICK Department of Computer Science, DePaul University, Chicago, USA [email protected] Accepted 6 January 2004 ABSTRACT We show that there are infinitely many two component links in S 3 whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting. Keywords: Heegaard splittings; weakly reducible; irreducible; primitive meridian. Mathematics Subject Classification 2000: 57N25

1. Introduction A well known result of Casson and Gordon (see [1, Theorem 3.1]) states that if M is a closed orientable manifold and (V1 , V2 ) is an irreducible but weakly reducible Heegaard splitting of M then M contains an essential surface of positive genus. This theorem has been used extensively and it was a natural thing to expect an extension of it to manifolds with boundary. The fact that the statement of [1, Theorem 3.1] does not extend as is to manifolds with boundary was first shown by the second author in [10]. These examples involve manifolds which are complements of three component links in S 3 but do not extend to manifolds with fewer boundary components. In this paper we show that the construction method provided in [1, Theorem 3.1] fails for manifolds with two boundary components. The proof was considerably more difficult than for manifolds with three or more boundary components. We do this by showing, in Theorems 3.2 and 3.4, the claim for two different infinite families of knot complements in S 3 . We thus obtain: 829


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Theorem 1.1. There are manifolds with non-empty boundary consisting of two tori which have a weakly reducible irreducible Heegaard splitting yet all essential surfaces obtained by compressing a maximal weakly reducing system of disks yields a boundary parallel surface. The question remains open for manifolds with a single boundary component. Since the only way an extension of the Casson and Gordon theorem to manifolds with boundary can fail is when the weakly reducible Heegaard splitting in question is a boundary stabilization (see [7]) the following theorem is in the “positive” direction: Theorem 4.6. Let (V1 , V2 ) be a Heegaard splitting for a manifold M with a single boundary component which has genus reducing surgery and is arc connected, then a ∂-stabilization is a stabilization. The term genus reducing surgery (see Definition 4.1) means that the Heegaard splitting becomes stabilized after attaching a 2-handle (Dehn filling in the case of a manifold with torus boundary). Arc connected is a property of a Heegaard surface defined in Sec. 4. Given a candidate for a counter example, i.e. a weakly reducible non-minimal genus Heegaard splitting, the difficult part is to show that it is in fact irreducible (non-stabilized). A straightforward example for the failure of an extension of the Casson and Gordon theorem to manifolds with three or more boundary components is given in [10, Example 6.1]:

τ2

τ1

Fig. 1.1.

Let M = (pair of pants)×S 1 . M is homeomorphic to the exterior of the three component chain shown in Fig. 1.1. M has a genus three Heegaard surface Σ from the tunnel system consisting of the two tunnels τ1 ∪ τ2 . The Heegaard surface Σ cuts M into a handlebody of genus three and a compression body with three torus boundary components. However this splitting is weakly reducible. The cocore of τ1 , say, is disjoint from a compressing disk for the handlebody which runs along τ2 only. It is easy to see that Σ is also irreducible, because it is minimal genus for the partition of the boundary components given by Σ. A reduction/destabilization of a Heegaard surface produces a Heegaard surface of lesser genus but with the same partition of the boundary components. In this case it is impossible to have a


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Heegaard surface of genus less than three which splits M into a handlebody and a compression body with three torus boundary components. Since the complement of the link is homeomorphic to a (pair of pants) × S 1 it contains no closed essential surfaces. Note that a minimal genus splitting for (pair of pants)×S 1 is of genus two with one compression body containing two boundary components and the other one boundary component. In Fig. 1 above we exhibit a weakly reducible yet irreducible splitting of the three component chain. Remark. Possible Heegaard splitting candidates for a counter example cannot be γ-primitive in the sense of [7]. However Heegaard splittings which are γ-primitive are very common. Hence possible counter examples to the remaining cases are also hard to find. 2. Preliminaries In this paper it is assumed that all manifolds and surfaces will be orientable unless otherwise specified. A compression body V is a compact orientable and connected 3-manifold with a preferred boundary component ∂+ V and is obtained from a collar of ∂+ V by attaching 2-handles and 3-handles, so that the connected components of ∂− V = ∂V − ∂+ V are all distinct from S 2 . The extreme cases, where V is a handlebody i.e. ∂− V = ∅, or where V = ∂+ V × I, are allowed. Alternatively we can think of V as obtained from (∂− V ) × I by attaching 1-handles to (∂− V ) × {1}. An annulus in a compression body will be called a vertical (or a spanning annulus) if it has its boundary components on different boundary components of the compression body. Given a manifold M 3 a Heegaard splitting (V1 , V2 ) for M is a decomposition M = V1 ∪ V2 into two compression bodies so that V1 ∩ V2 = ∂V1 = ∂V2 = Σ. The surface Σ will be call the Heegaard splitting surface. A Heegaard splitting (V1 , V2 ) for a manifold M will be called reducible if there are essential disks D1 ⊂ V1 and D2 ⊂ V2 so that ∂D1 = ∂D2 ⊂ Σ. A Heegaard splitting (V1 , V2 ) for a manifold M will be called stabilized if there are essential disks D1 ⊂ V1 and D2 ⊂ V2 so that ∂D1 ∩ ∂D2 = p ⊂ Σ, where p is a single point. For Heegaard splittings of genus bigger than one irreducible implies non-stabilized but if the manifold M is irreducible then a Heegaard splitting is irreducible if and only if it is non-stabilized. A Heegaard splitting (V1 , V2 ) for a manifold M will be called weakly reducible if there are disjoint essential disks D1 ⊂ V1 and D2 ⊂ V2 . Otherwise it will be called strongly irreducible. Let M be a 3-manifold which is homeomorphic to a (surface) × I. A Heegaard splitting (V1 , V2 ) of M will be called standard if it is homeomorphic to one of the following types: (I) V1 ∼ = (surface)×[ 21 , 1] and ∂+ V1 = ∂+ V2 = (surface)× [ 21 ]. = (surface)× [0, 21 ], V2 ∼ (II) If {p} ∈ (surface) is a point then for 0 < < 21 V1 ∼ = ((surface)×[0, ])∪(N (p)×I)∪((surface)×[1−, 1]) and V2 = cl(M −V1 ).


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Note that V2 is a regular neighborhood of a once punctured surface and hence is a handlebody and V1 is a compression body with one boundary component ∂+ of genus 2g and two boundary components ∂− of genus g, where g = genus (surface). In [11] it is proved that any non-stabilized Heegaard splitting of (surface) × I is homeomorphic to one of the above two types. A closed surface F ⊂ M will be called essential if it incompressible and nonboundary parallel. Given a closed (possibly disconnected) surface Σ ⊂ M and a system of pairwise disjoint non-parallel compressing disks ∆ for Σ define (as in [1]) Σ0 = σ(Σ, ∆) to P be the surface obtained from Σ by compressing along ∆. Let c(Σ) = i (1 − χ(Σi )), where the sum is taken over all components Σi of Σ which are not 2-spheres. The complexity of the system ∆ is defined to be: c(∆) = c(Σ) − c(Σ0 ) For a given Heegaard splitting surface Σ for M we will assume that a system of compressing disks ∆ = ∆1 ∪ ∆2 , where ∆i ⊂ Vi , satisfies: (a) ∆i 6= ∅ for both i = 1, 2. i.e. ∆ contains disks on both sides of Σ. (b) ∆ is maximal with respect to c(∆) over all systems ∆ satisfying (a). Let Σ∗ be the surface Σ0 less the 2-sphere components and the components which are contained in V1 or V2 . In [1, Theorem 3.1], restated below, Casson and Gordon prove: Theorem 2.1. If M is a closed 3-manifold which has a weakly reducible and irreducible Heegaard splitting then the surface Σ∗ is incompressible. In particular M is Haken. Definition 2.2. Given two manifolds M1 and M2 with respective Heegaard splittings (U11 , U12 ) and (U21 , U22 ), assume further that there are homeomorphic boundary components F1 ⊂ ∂− U11 and F2 ⊂ ∂− U21 . Denote the homeomorphism F1 → F2 by g. Let M be a manifold obtained by gluing F1 and F2 along the homeomorphism g. We can obtain a Heegaard splitting (V1 , V2 ) for M by a process called amalgamation as follows: Given a compression body U we can assume that it has the structure of ∂− U × I ∪ {1-handles}. Let hi be the homeomorphism N (∂− Ui1 ) → (∂− Ui1 ) × I and pi : (∂− Ui1 ) × I → ∂− Ui1 the projection into the first factor. Define an equivalence relation ≡ on M1 , M2 as follows: (1) If xi , yi ∈ N (Fi ) are points such that pi hi (xi ) = pi hi (yi ) then xi ≡ yi . (2) If x ∈ F1 , y ∈ F2 and g(x) = y, where g : F1 → F2 is the homeomorphism between the surfaces, then x ≡ y. Furthermore we can arrange that the attaching disks on F1 × {1}, (F2 × {1}) for the 1-handles in U11 , U21 respectively, have disjoint images in F1 (F2 ) and hence they


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do not get identified to each other. Now set M = (M1 ∪M2 )/ ≡, V1 = (U11 ∪U22 )/ ≡, V2 = (U12 ∪U21 )/ ≡. Note that V1 = U22 ∪N (F1 )∪{1-handles}∪(∂− U11 −F1 )×I and V2 = U12 ∪ N (F2 ) ∪ (1-handles) ∪ (∂−U21 − F1 ) × I so that V1 and V2 are compression bodies defining a Heegaard splitting (V1 , V2 ) for M . The Heegaard splitting (V1 , V2 ) of M is called the amalgamation of the Heegaard splittings (U11 , U12 ) of M1 and (U21 , U22 ) of M2 along F1 and F2 . The process of amalgamation reconstructs the original Heegaard splitting (V1 , V2 ) of M from the Heegaard splittings induced on the components Ni of M −Σ∗ . We depict the process of amalgamation in the following figure. U12

Σ(M1 ) ∂_M1 ∂_M2 Σ(M2)

Σ(M)

U22 Fig. 2.1.

Given a manifold M with boundary components ∂M 1 , . . . , ∂M k of corresponding genus g 1 , . . . , g k and a Heegaard splitting (V1 , V2 ) for M of genus g we can always obtain a new Heegaard splitting (U1i , U2i ) of genus g + g i , i = 1, . . . , k, by gluing a ∂M i ×I to the ∂M i boundary component and then amalgamating the standard Heegaard splitting of genus 2g i of ∂M i × I with the given Heegaard splitting (V1 , V2 ) of M (as indicated in Fig. 2.1). Definition 2.3. The construction above will be called boundary stabilization on the ith boundary component. If there is a single boundary component or no ambiguity we can just use boundary stabilization or ∂-stabilization. Remark 2.4. (1) A Heegaard splitting which is a boundary stabilization is weakly reducible. The cocore disk of the single 1-handle in the compression body part of the Heegaard splitting of ∂M i × I is clearly disjoint from any cocore disk in any 1-handle of the compression body part of the Heegaard splitting of M not containing ∂M i . (2) ∂-stabilizing twice in the same boundary component yields a stabilized Heegaard splitting. We can first amalgamate the two successive Heegaard splittings of ∂M i ×I to obtain a Heegaard splitting of genus 2g i +1, where g i is the genus of ∂M i , which is stabilized by [11]. (3) Every Heegaard surface induces a partition of the boundary components of the manifold, those that are contained in V1 and those contained in V2 . Moreover,


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it follows from Item (5) that for every partition of boundary components there is an irreducible Heegaard splitting inducing that partition. (4) If the ith boundary component ∂M i belonged to, say, V1 then after ∂-stabilizing along ∂M i the “new” ith boundary component ( the other boundary components are not affected by this operation) now belongs to V2 . In other words ∂stabilizing changes the partition by moving the boundary component on which we ∂-stabilized from one compression body to the other. (5) If a Heegaard splitting is stabilized then it contains a pair of essential disks which intersect in a single point i.e. a stabilizing disk-disk pair. Destabilizing the Heegaard splitting is removing a regular neighborhood of one of the stabilizing disks from one compression body and attaching it as a 2-handle to the other. This operation does not change the partition of the boundary components between the two compression bodies in contrast to ∂-stabilization. Finally we need the following definitions (see [2]): Definition 2.5. A link L ⊂ S 3 is tunnel number one (denoted by t(L) = 1) if S 3 − N (L) has a genus two Heegaard splitting with one component a compression body and the other a handlebody. Remark 2.6. Note that for knots K ⊂ M 3 we have that g(M −N (K)) = t(K)+1, where g() denotes the genus. However for links L ⊂ M 3 there can be a difference between the genus of M − N (K) and t(L) + 1. This can be seen in [10, Example 6.2] and [3, Sec. 5]. The number t(L) + 1 will always be an upper bound on the genus of M − N (K). Definition 2.7. A link L ⊂ S 3 is tangle composite if the pair (S 3 , L) can be decomposed as (S 3 , L) = (B1 , T1 ) ∪ (B2 , T2 ), where (Bi , Ti ), i = {1, 2}, are essential tangles, i.e. Bi is a 3-ball and Ti is a collection of n arcs properly embedded in Bi . Furthermore ∂Bi − N (Ti ) are incompressible and ∂-incompressible. The union identifies ∂(B1 , T1 ) with ∂(B2 , T2 ). Definition 2.8. An n-string Hopf tangle is a tangle as depicted in Fig. 2.2.

...

Fig. 2.2.

An n-string Hopf tangle.


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3. Examples 0

α 3 Example 3.1. Let K1 ( α β ), K2 ( β 0 ) be two non-trivial 2-bridge knots in S and 0

α L = K1 ( α β ) ∪∗ K2 ( β 0 ) be the two components linked as in Fig. 3.1.

Fig. 3.1.

The link L is a couple of two bridge knots.

Theorem 3.2. The manifold M = S 3 −N (L) has a weakly reducible and irreducible Heegaard splitting of genus three such that any component of any Σ ∗ is boundary parallel into one of the two torus boundary components of M . Proof. The genus of the manifold M = S 3 − N (L) is less than or equal to two as can be seen by attaching the top standard unknotting tunnel to K1 ( αβ ) and the 0

bottom unknotting tunnel to K2 ( α β 0 ). This Heegaard splitting (V1 , V2 ), is composed of two compression bodies each with a single ∂− boundary component, which is a 2-torus (see Fig. 3.2). The genus cannot be one as this would imply that the Heegaard surface will be a torus that is parallel into both link components. But this can only happen in the complement of the Hopf link, which is a contradiction.

Fig. 3.2.

The complement of L has Heegaard genus two.


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Choose a boundary component, say, ∂M 1 = ∂N (K1 ( αβ )) of M and ∂-stabilize along that component. By Remarks 2.4(1) and (4), since ∂M 1 = T 2 , we obtain a new Heegaard splitting (W1 , W2 ) of genus three for M which is weakly reducible and is composed of one handlebody and one compression body with two ∂− tori boundary components. Suppose now that (W1 , W2 ) is stabilized. Since destabilizing a Heegaard splitting does not change the partitioning of the boundary components with respect to the two compression bodies (Remark 2.4(5)), the new genus two Heegaard splitting (U1 , U2 ) is composed of a handlebody U1 and a compression body U2 such that ∂− U2 = T 2 ∪ T 2 . This would imply that the link L is a tunnel number one link. In [2, Theorem 1.5] Gordon and Reid prove the following theorem: Theorem. A tunnel number one link in S 3 is tangle composite if and only if it has an n-string Hopf summand for some n. Clearly the link L does not have an n-string Hopf summand as both its components are knotted while one component of an n-string Hopf summand is unknotted. Consider now the decomposition of (S 3 , L) into two tangles (Bi , Ti ), i = 1, 2, as indicated in Fig. 3.3.

Fig. 3.3.

A tangle decomposition of the couple.

The two tangles are essential by [12, Lemma 3.1]. Hence the only option left is that L is not a tunnel number one link. We thus have a contradiction and the Heegaard splitting (W1 , W2 ) is irreducible (non-stabilized). The two are equivalent in this case as [12, Lemma 3.1] also shows that S 3 − N (L) is irreducible and that L is a non-split link. We can now apply the construction of [1] as in Sec. 2 and compress the Heegaard surface Σ = ∂+ W1 = ∂+ W2 along a maximal collection ∆ of weakly reducing disks to obtain a surface Σ∗ . The disk system ∆ can contain at most two disks. If |∆| = 3 then Σ∗ will be an essential S 2 which is a contradiction as L is a non-split link. If |∆| = 2 then Σ∗ = T 2 .


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First note that Bi −N (L) is a genus two handlebody for i = 1, 2. This is because Bi − N (L) is homeomorphic to S 3 − Ki − τ , where τ is one of the “dual” tunnels for the 2-bridge knot Ki (see [5]). Since Bi − N (L) is a genus 2 handlebody it does not contain an incompressible torus. Moreover, if a handlebody (other than a solid torus) contains an incompressible annulus whose boundary components cobound an annulus in the boundary of the handlebody, then either the incompressible annulus is peripheral into the boundary annulus, or the boundary of the handlebody minus the boundary annulus is compressible in the handlebody. Second, note that L is not a non-trivial connected sum: If it were it would contain an essential decomposing annulus A with meridional slope on one of the knots, say K1 . Choose such an annulus A that intersects the essential four punctured sphere ˆ for such Sˆ = ∂B1 − N (L) minimally. The annulus A cannot be disjoint from S, an annulus is contained in a genus two handlebody Bi − N (L) and has parallel boundary curves on (∂Bi − N (L)). By the previous remarks A would either be peripheral, a contradiction, or ∂(Bi − N (L)) is compressible in its complement contradicting the fact that the tangles are essential. But if A met Sˆ then it would do so in a collection of curves which are parallel on A. The curve closest to a component of ∂A is a curve on Sˆ which necessarily separates the punctures by K1 and not those of K2 as A does not intersect the knot K2 . Hence that curve and the component of ∂A are parallel in (∂Bi − N (L)) defining a sub-annulus A0 of A that is embedded in Bi − N (L). Again, the fact that the tangles are essential implies that the sub-annulus A0 is peripheral. An isotopy will reduce the intersection of A and Sˆ in contradiction to the choice of A. Assume now, in contradiction, that M contains an essential torus. Choose such an essential torus T that meets Sˆ minimally. The surfaces T and Sˆ are not disjoint because Bi − N (L) is a handlebody. Furthermore no curve in the intersection of T ˆ This would imply and Sˆ bounds a disk in ∂Bi containing just one puncture of S. that T is meridionally compressible hence L is a non-trivial connected sum or T peripheral. So each curve of intersection on Sˆ separates two punctures from the other two and hence they are all parallel on Sˆ ⊂ Bi − N (L). It follows that each annular component of T ∩ Bi has parallel boundary curves on the handlebody Bi − N (L) and is therefore peripheral to an annulus in the level sphere. An isotopy will reduce the number of intersections, contradicting our assumption that the intersection is minimal. Example 3.3. Let Lm = Km ∪O ⊂ S 3 , be the link depicted in Fig. 3.4, where Km is the Morimoto Sakuma Yokota knot studied in [8]. It is the (7, 17)-torus knot with two adjacent strands twisted around each other to add r = 10m − 4 crossings. Let t be an unknotting tunnel for Km which is an essential arc in the obvious essential annulus in S 3 − N (Km ). We can isotope Km ∪ t to lie on a 2-torus T standardly embedded in S 3 dividing it into two solid tori V and V 0 . Choose O as a core of, say, V 0 . The knot Km has the property that it is not µ-primitive i.e. it has no minimal


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Km

.......

(7,17) torus knot

r=10m - 4 crossings

O Fig. 3.4.

A link formed from the Morimoto knot Km .

genus Heegaard splitting (V1 , V2 ) where the meridian µ ⊂ ∂− V1 is isotopic to a primitive curve in ∂+ V2 (see Definition 4.4 and also [7]). The manifold M = S 3 − N (Lm ) has a genus two Heegaard splitting (V1 , V2 ), where the components are two compression bodies, which is obtained by adding the unknotting tunnel t to Km . Theorem 3.4. The manifold M = S 3 − N (Lm ) has a weakly reducible and irreducible (non-stabilized) Heegaard splitting of genus three such that any component of any Σ∗ is boundary parallel into one of the two tori boundary components of M . Proof. The Heegaard splitting (V1 , V2 ) consists of two genus two compression bodies each with ∂− Vi = T 2 . Do a ∂-stabilization on the O component of Lm . We obtain a weakly reducible Heegaard splitting (U1 , U2 ) of genus three for S 3 − N (Lm ) with both boundary components contained in a single compression body (see Remark 2.4(1), (4)). The link Lm is not a split link since the linking number of the two components is seven. Hence the manifold S 3 − N (Lm ) is irreducible. Assume that (U1 , U2 ) is stabilized. After reducing we obtain a genus two Heegaard splitting (W1 , W2 ) with a single compression body W1 such that ∂− W1 = T 2 ∪ T 2 (see Remark 2.4(5)). This implies that Lm is a tunnel number one link with an unknotting tunnel t1 which connects the two link components. However since the component O of Lm is unknotted we can think of Km as a knot in the solid torus


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V = S 3 − N (O). The complement of N (Km ∪ t1 ) in V is the genus two handlebody W2 . We have thus obtained a genus two Heegaard splitting for V composed of one handlebody which is equal to W2 less a collar, and one compression body W ∗ = ∂V × I ∪ N (Km ∪ t1 ). This compression body W ∗ has a unique, up to isotopy, non-separating disk D: If not, let E be another such disk not parallel to D. Since it is essential and non-separating and W ∗ is of genus two E ∩ D 6= ∅. Choose E that minimizes |E ∩ D| and consider an outermost arc of intersection in D. This yields a boundary compression of E into two disks which are essential. By the minimality of E both are separating. But then E is separating as well. All Heegaard splittings of a solid torus are standard by [1]. The well defined, non-separating disk D is the meridian of the knot Km , and also the cocore of the standard splitting of genus two of the solid torus. Because of this last point, there is a disk in the complement which meets D exactly once. Therefore Km is µprimitive. This is a contradiction, as Km is not µ-primitive because t(Km #Km ) = t(Km ) + t(Km ) + 1 (see [8]). As the link Lm is not a split link, S 3 − N (Lm ) does not contain an essential S 2 and a maximal weakly reducing disk system ∆ contains exactly two disks. Since the genus of (U1 , U2 ) is three it follows that Σ∗ must be a 2-torus T . Assume therefore that S 3 − N (Lm ) contains such an incompressible torus T . The knot Km is contained in the solid torus V with meridional disk denoted by D0 . Moreover, Km is a braid in V with respect to O as an axis, so that V − D0 − N (Km ) is a product (punctured disk) × I. When T is chosen to minimize the intersection of T ∩ D 0 then T − D0 is decomposed into a collection of vertical annuli in this product structure. Consider an innermost curve on the punctured disk D0 − N (Km ). An annulus with this curve as boundary has its other boundary curve also an innermost curve bounding a sub-disk of D 0 with the same number of punctures. By connectivity all curves of intersection are innermost and must bound sub-disks of D0 with the same number of punctures. As T ∩ D 0 cannot contain curves bounding sub-disks without any punctures, since T is incompressible, it follows that T separates ∂V and Km . Hence every puncture is bounded by such a curve. This partitions the punctures into sets each with the same number of punctures, thus factoring the number of punctures. Depending on our choice of D0 , it could be punctured either seven or seventeen times. Since both numbers are prime, there are either curves of intersection bounding sub-disks containing all punctures, or curves each bounding a sub-disk containing a single puncture. It follows that T is peripheral, to O or to Km , respectively (see Fig. 3.5). It follows that any incompressible torus T in S 3 − N (Lm ) is boundary parallel and the proof is complete. Note that the same arguments, of the above proof, prove the following more general theorem:


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T

D

Fig. 3.5.

Km

The meridional disk D.

Theorem 3.5. Let K ⊂ S 3 be a tunnel number one knot which is not µ-primitive. Let O be an unknot that is a core of the genus two handlebody in a minimal genus Heegaard splitting of S 3 − N (K). Then any boundary stabilization of a genus two Heegaard splitting of S 3 − N (L), where L is the link K ∪ O, is non-stabilized. Remark 3.6. In particular if L, as above, is a hyperbolic link then S 3 − N (L) is a counter example for a possible extension of the Casson and Gordon theorem to manifolds with two boundary components. 4. Manifolds with a Single Boundary Component In this section we state and prove a partial result in the case of 3-manifolds with a single boundary component. The situation in general is not yet completely understood. Definition 4.1. A Heegaard splitting (V1 , V2 ) for (M, ∂M ) is said to have a genus reducing surgery or g.r.s. if there is a curve γ on ∂M such that after doing 2-surgery on γ and adding 3-balls to S 2 boundary components, if need be, we obtain a manifold pair (M 0 , ∂M 0 ) and the induced Heegaard splitting (V10 , V20 ) = (V10 (γ), V20 (γ)) on (M 0 , ∂M 0 ) is stabilized. Remark 4.2. Any Heegaard splitting for a knot complement in S 3 has a genus reducing surgery along a meridional curve. If a Heegaard splitting for a manifold with a single boundary component has a g.r.s. then there is a disk D in the handlebody component of the Heegaard splitting and a punctured disk P in the compression body so that D ∩ P is a single point. In this case we say that we have a planar surface disk destabilizing pair or a (P, D)-pair. We say that an element x in a free group Fn is primitive if it belongs to some basis for Fn . A curve on a handlebody H is primitive if it represents a primitive


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element in the free group π1 (H). An annulus A properly embedded in H is primitive if its core curve is primitive. Note that a curve on a handlebody is primitive if and only if there is an essential disk in the handlebody intersecting the curve in a single point. Definition 4.3. Let M be a 3-manifold with incompressible boundary components ∂M 1 , . . . , ∂M k . Let γ ⊂ ∂M i be an essential simple closed curve. A Heegaard splitting (V1 , V2 ) of M will be called γ-primitive if there is an annulus A in V1 or V2 , say V1 , with γ as one boundary component of A and the other a curve on the Heegaard surface Σ which intersects an essential disk of V2 in a single point. In this case we have g.r.s. and the punctured disk P of Remark 4.2 is an annulus A. We also say that we have an annulus disk destabilizing pair or (A, D)-pair. Definition 4.4. Let M be a 3-manifold with incompressible boundary components ∂M 1 , . . . , ∂M k . The manifold M will be called boundary primitive or (∂-primitive) if for each Heegaard splitting of M and for each boundary component ∂M i there is some curve γ ⊂ ∂M i for which the Heegaard splitting is γ-primitive. In particular if M = S 3 − N (K) where K ⊂ S 3 is a knot and γ = µ is a meridian curve we will say that the Heegaard splitting is µ-primitive and that K is µ-primitive if all its Heegaard splittings are µ-primitive. Note that a Heegaard splitting obtained by boundary stabilization on a boundary component is γ-primitive for every curve γ on that boundary component. We can always think of a compression body with a single ∂− component as a (surface) × I attached along a disk to a handlebody. Hence if we have a (P, D)-pair in such a Heegaard splitting we can assume that the punctured disk P is a collection of vertical annuli in the (surface) × I component and some disks in the handlebody part which are connected together by bands. Definition 4.5. A Heegaard splitting (V1 , V2 ) with a g.r.s. of a manifold (giving a (P, D)-pair) with a single boundary component in V1 is arc connected if there is an arc α in ∂+ V1 ∩ (∂− V1 × I), for some product structure on V1 , such that α intersects all the vertical annuli of P and α ∩ D = ∅. Theorem 4.6. Let (V1 , V2 ) be a Heegaard splitting for a manifold M with a single boundary component which has a genus reducing surgery and is arc connected, then a ∂-stabilization is a stabilization. Proof. Since (V1 , V2 ) has a genus reducing surgery it has a (P, D)-pair. As P is a planar surface in a compression body we can boundary compress along the bands to get a collection {∪Ai } of vertical annuli. As (V1 , V2 ) is arc connected there is an arc α ⊂ ∂+ V1 so that α ∩ ∂+ Ai is not empty for each of the annuli (∂+ Ai is the boundary component of Ai which is on ∂+ V1 ). However α ∩ D is empty. Now set D0 = N∂+ V1 (α) and boundary stabilize by drilling the “shaft” D 0 × I. This boundary stabilization removes at least one essential arc from each of the annuli


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Ai so that P − D0 × I is an essential disk D1 . This operation has no effect on the single point of intersection between P and D. Now we have an essential disk D1 which meets the disk D from the (P, D)-pair in a single point. Hence the boundary stabilized Heegaard splitting is a stabilization. Corollary 4.7. Let K ⊂ S 3 be a knot such that all of its strongly irreducible splittings are γ-primitive. If S 3 − N (K) has a weakly reducible and irreducible Heegaard splitting then it contains an essential surface. Remark 4.8. It follows that if one wishes to find a counter example to a possible extension of [1, Theorem 3.1] to manifolds which are knot complements in S 3 one is forced to consider knots which are not µ-primitive. This class of knots which we call fiendish knots is very elusive. The first proof that they exist is in [2] where it was proved that infinite families of knots which are tunnel number super additive. Being non-µ-primitive is a necessary condition for being tunnel number super additive. Actual examples of knots which are tunnel number super additive were exhibited in [8]. Recently Eudave-Munoz [9] found infinite families of knot which are non-µprimitive. The smallest such knot is the mutated (7, 17)-torus knot of [8] which has 107 crossings. Acknowledgment The first author was supported by The Fund for Promoting Research at the Technion, grant 100-127 and the Technion VRP fund, grant 100-127. References [1] A. Casson and C. Gordon, Reducing Heegaard splittings, Topol. Appl. 27 (1987) 275–283. [2] C. Gordon and A. Reid, Tangle decompositions of tunnel number one knots and links, J. Knot Theory Ramifications 4 (1995) 389–409. [3] T. Kobayashi, Scharlemann–Thompson untelescoping of Heegaard splittings is finer than Casson–Gordon’s, preprint. [4] T. Kobayashi, Heegaard splittings of exteriors of two bridge knots, Geom. Topol. 5 (2001) 609–650. [5] T. Kobayashi, Classification of unknotting tunnels for two bridge knots, in Proceedings of Kirbyfest, Geometry and Topology Monographs, Vol. 2 (1999), pp. 259–290. [6] M. Lustig and Y. Moriah, Closed incompressible surfaces in complements of wide knots and links, Topol. Appl. 92 (1999) 1–13. [7] Y. Moriah, On boundary primitive manifolds and a theorem of Casson–Gordon, Topol. Appl. 125(3) (2002) 571–579. [8] K. Morimoto, M. Sakuma and Y. Yokota, Examples of tunnel number one knots which have the property “1 + 1 = 3”, Math. Proc. Cambridge Phil. Soc. 199 (1996) 113–118. [9] M. Eudave-Munoz, (1,1)-knots and incompressible surfaces, preprint.


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[10] E. Sedgwick, Genus two 3-manifolds are built from handle number one pieces, Algebr. Geom. Topol. 1 (2001) 763–790. [11] M. Scharlemann and A. Thompson, Heegaard splittings of (surfaces) × I are standard, Math. Ann. 95 (1993) 549–564. [12] Y.-Q. Wu, The classification of non-simple algebraic tangles, Math. Ann. 304 (1996) 457–480.