The Morse-Novikov number of knots under connected sum and cabling
TTHE MORSE-NOVIKOV NUMBER OF KNOTS UNDER CONNECTED SUM ANDCABLING.
KENNETH L. BAKER
Abstract.
We show the Morse-Novikov number of knots in S is additive under connected sum and un-changed by cabling. Given an oriented link L in S , the Morse-Novikov number of L is the count MN ( L ) of the minimumnumber of critical points among regular Morse functions f : S − L → S , see [VPR01] and [Paj06, Definition14.6.2]. Pajitnov attributes to M. Boileau and C. Weber the question of whether if the Morse-Novikovnumber is additive on the connected sum of oriented knots, see the beginning of [Paj10, Section 5] and theend of [Paj06, Section 14.6.2]. We show that it is. Theorem 1.1.
The Morse-Novikov number is additive: If K = K a K b is a connected sum of two orientedknots K a and K b in S , then MN ( K ) = MN ( K a ) + MN ( K b ) . Instead of working with circle-valued Morse functions directly, we use the handle-theoretic interpretationof the Morse-Novikov number presented in [God06, §
3] and [GP05, §
2] that enables the use of techniquesfrom the theory of Heegaard splittings. This approach is rooted in Goda’s work on handle numbers ofsutured manifolds and Seifert surfaces [God92,God93] and Manjarrez-Gutierrez’s work on generalized circularHeegaard splittings and circular thin decompositions [MG09]. Notably, in [MG13, Theorem 1.1] Manjarrez-Gutierrez uses this approach to prove Theorem 1.1 for the special class of a-small knots (i.e. knots with noclosed essential surface disjoint from a Seifert surface) by using a key proposition about the positioning of thesumming annulus for a connected sum with respect to a circular (locally) thin decompositions of the knotexterior which she established with Eudave-Munoz [EMnMG12, Proposition 5.1]. In essence, we manageto avoid this a-small hypothesis in our proof of Theorem 1.1 by paying closer attention to the behavior ofcounts of handles in generalized circular Heegaard splittings under the operations of weak reductions andamalgamations.In preparation for [MG13, Theorem 1.1], Manjarrez-Gutierrez shows that the handle number of an a-smallknot is realized by the handle number of an incompressible Seifert surface, [MG13, Theorem 4.3]. On ourway to Theorem 1.1 we prove the analogous Lemma 1.2 below which removes the a-small hypothesis. Our handle number definitions are given in Definition 2.23.
Lemma 1.2.
A knot K ⊂ S has an incompressible Seifert surface R such that h ( K ) = h ( R ) . Finally, we observe that [EMnMG12, Proposition 5.1] directly generalizes to address cabling annuli. Con-sequently, our proof of Theorem 1.1 adapts to show that handle number is unaffected by cabling.
Theorem 1.3.
Let K p,q be the ( p, q ) –cable of the knot K for coprime integers p, q with p > . Then h ( K p,q ) = h ( K ) and hence MN ( K p,q ) = MN ( K ) . For the ease of exposition, we content ourselves with focusing upon knots in S . However, Lemma 1.2,Theorem 1.1, and Theorem 1.3 can be immediately generalized to null-homologous knots in rational homologyspheres. With a little more work they should also generalize to rationally null-homologous knots in otherorientable 3–manifolds. Mathematics Subject Classification.
Key words and phrases.
Morse-Novikov, handle number, connected sum, cabling, Heegaard splittings, weak reduction,amalgamation. a r X i v : . [ m a t h . G T ] J a n Let us quickly sketch the proofs of Lemma 1.2 and Theorem 1.1 for readersalready familiar with the notions of (circular) generalized Heegaard splittings. Their full proofs are given in §
3. As the proof of Theorem 1.3 is quite similar to that of Theorem 1.1, we wait to address it in § §
4. Prior to the two sketches, there are a few things worthclarifying now which we will address more fully in § M (with positive and negative subsurfaces R + and R − of ∂M , with annular sutures betweentheir boundaries, and possibly with toroidal sutures) and a disjoint pair of properly embedded (possiblydisconnected) “thin” surface R and “thick” surface S that decompose M into compression bodies. Thepositive boundaries of connected compression bodies are components of S , while the negative boundariesare unions of components of R ∪ R + ∪ R − (satisfying some conditions on R + and R − ). We also frequentlysuppress the term circular as it is implied when M has a toroidal suture.The handle number h ( W ) of a compression body W is the minimum number of 0– and 1– handles used in itsconstruction. We define its handle index j ( W ) to be the number of 1–handles minus the number of 0–handlesused in its construction. (Dually, it is the number of 2–handles minus the number of 3–handles.) This turnsout to be half of the handle index J introduced by Scharlemann-Schultens [SS00, SS01]. Importantly, thehandle index j agrees with the handle number h when the compression body has no handlebody component.We extend both the handle index and handle number to generalized Heegaard splittings by summing over thecompression bodies in the decomposition. (Our use of handle number differs from Goda’s at this point.) Thedriving observation is that, because weak reductions and amalgamations of generalized Heegaard splittingsneither introduce nor cancel 1– & 2–handle pairs, the handle index is unchanged by these operations. Sketch of proof of Lemma 1.2.
Consider a circular Heegaard splitting (
M, R, S ) of the exterior M of theknot K with compressible Seifert surface R where h ( K ) = h ( R ) = h ( M, R, S ). Since the splitting containsno handlebodies, h ( M, R, S ) = j ( M, R, S ). Since R is compressible, the splitting is weakly reducible. Amaximal weak reduction along S produces a generalized Heegaard splitting ( M, R ∪ R , S ∪ S ) where R = R and R is a potentially disconnected surface that contains an incompressible Seifert surface R . Thenamalgamations along R ∪ R − R produce a circular Heegaard splitting ( M, R , S (cid:48)(cid:48) ) without handlebodies.Thus j ( M, R, S ) = j ( M, R , S (cid:48)(cid:48) ) = h ( M, R , S (cid:48)(cid:48) ) ≥ h ( R ) ≥ h ( K ). Hence h ( R ) = h ( K ). See Figure 1 for aschematic of the weak reduction and amalgamations. (cid:3) Sketch of proof of Theorem 1.1.
We show h ( K a K b ) ≥ h ( K a ) + h ( K b ) as the other inequality is straight-forward. Let M be the exterior of K and let Q be the annulus that decomposes M into the exteriors M a and M b of the constituent knots K a and K b .By Lemma 1.2, there is an incompressible Seifert surface R for K = K a K b such that h ( K ) = h ( R ).Hence there is a circular Heegaard splitting ( M, R, S ) realizing this handle number. A maximal iteratedweak reduction of this splitting yields a locally thin generalized Heegaard splitting ( M, R , S ) in which R isa component of R . Thus h ( K ) = h ( R ) = h ( M, R, S ) = j ( M, R, S ) = j ( M, R , S ) . By [EMnMG12, Proposition 5.1], since ( M, R , S ) is locally thin, we may isotop Q so that any intersectionwith a compression body of the splitting is a product disk. Then Q chops this splitting of M into splittings( M i , R i , S i ) of M i for i = a, b . Furthermore j ( M, R , S ) = j ( M a , R a , S a ) + j ( M b , R b , S b ) . For each i = a, b , a sequence of amalgamations of ( M i , R i , S i ) produces a circular Heegaard splitting( M i , R i , S i ) with no handlebodies where R i is a Seifert surface for K i . Hence j ( M i , R i , S i ) = j ( M i , R i , S i ) = h ( M i , R i , S i ) ≥ h ( R i ) ≥ h ( K i )and the result follows. (cid:3) — 1.2 Questions — Lemma 1.2 prompts the following question.
Question 1.4.
Is the handle number of a knot always realized by a minimal genus Seifert surface? hile Goda shows this is so for all small crossing knots, he also reveals the challenge with addressingthis question: there are knots with multiple minimal genus Seifert surfaces that do not all realize the handlenumber of the knot. See [God93]. — • —One may also wonder whether Theorem 1.1 might be proven more simply, albeit possibly more indirectly,by expressing the Morse-Novikov number in terms of other established knot invariants. Question 1.5.
Can the Morse-Novikov number of a knot be expressed in terms of other established knotinvariants?
While additivity under connected sum is not an uncommon property among knot invariants, the detectionof fiberedness is rather exceptional. For example, while the log of the leading coefficient of the Alexanderpolynomial is additive under connected sums of knots, it fails to detect fibered knots. (Indeed, the Alexan-der polynomial is multiplicative under connected sums and many non-fibered knots have monic Alexanderpolynomials.) Nevertheless, the log of the rank of the knot Floer Homology of a knot K in the highestnon-zero grading, LR ( K ) = log rk (cid:91) HFK(
K, g ( K )), is both additive on connected sums [OS04, Theorem 7.1]and equals zero precisely for fibered knots [Ghi08, Ni07]. However, LR is distinct from MN ; neither is afunction of the other. For alternating knots, rk (cid:91) HFK(
K, g ( K )) equals the coefficient of the maximal degreeterm of the Alexander polynomial [OS03]. So LR varies greatly among the small crossing knots. However allnon-fibered knots of crossing number at most 10 have MN = 2 [God93, § MN = 2 via [God92] while LR increases with their twisting (as can be calculated fromtheir Alexander polynomials since they are alternating knots). Question 1.6.
What knot invariants detect fibered knots and are additive under connected sum? Which arealso unaffected by cabling?
See [TT18] for related work on knot invariants that detect the unknot and are additive under connectedsum. Note that MN does not detect the unknot as it is 0 for all fibered knots.— • —As a connected sums and cables are special kinds of satellite knots, one may wonder how handle numberbehaves under more general satellite operations. We refer the reader to § Question 1.7.
Let P be a pattern with non-zero winding number, and let K be a knot in S . Does h ( P ( K )) = h ( P ) + h ( K ) ? In Lemma 4.4 we show h ( P ( K )) ≤ h ( P ) + h ( K ). Part of the difficulty in establishing the equality ingeneral is dealing with how an essential torus may intersect a (locally thin circular generalized) Heegaardsplitting. For starters, see [SS00] and [Tho16]. We suspect Question 1.7 has a negative answer for certainknots and patterns.Note that in § h ( P ) and h ( K ) are probably too restrictive for a satisfactory understanding of h ( P ( K )) andQuestion 1.7 in general without constraints on winding number. — 1.3 Acknowledgements — We thank Andrei Pajitnov for introducing this problem to us duringhis visit to the University of Miami in Spring 2019. We thank Jennifer Schultens for her correspondence andboth Nikolai Saveliev and Chris Scaduto for prompting Question 1.5 and the comparison with knot Floerhomology. We also thank Scott Taylor for conversations and valuable commentary on an earlier draft. Thiswork was partially supported by a grant from the Simons Foundation (
Here we recall fundamental notions in the theory of Heegaard splittings and its generalizations. As thereis a bit of variation in the literature, this allows us to also establish our terminology and notation. We referthe reader to [CG87,God93,God92,God06] for the basic elements of our approach to compression bodies andHeegaard splittings and to [ST94,Sch03,SSS16,MG09,EMnMG12] for the core ideas of generalized Heegaardsplittings, circular Heegaard splittings, and circular generalized Heegaard splittings. For our purposes in this rticle, we take care to clarify the operations of weak reductions and amalgamations and refer the readerto [Sch93, Sch04, MG13, Lac08] for further discussions of these operations.There are a few things to keep in mind when reading the literature. Some authors require compressionbodies to be connected. We allow them to be disconnected in general but primarily give attention toconnected compression bodies. Some authors view compression bodies as cobordisms between closed surfaces.We however need them to be cobordisms rel- ∂ between surfaces that may have boundary. To a great extentthese distinctions don’t impact the results, though when they do the appropriate modification is typicallystraightforward.Thoughout we will restrict ourselves to only considering irreducible manifolds so that any embeddedsphere bounds a 3–ball. For notation, the result Y \ X of chopping a manifold Y along a submanifold X isthe closure of Y − X in the path metric. — 2.1 Compression bodies —Definition 2.1 (Sutured manifolds) . Introduced by Gabai [Gab83], a sutured manifold is a compact oriented3–manifold M together with a disjoint pair of subsurfaces R + and R − of ∂M such that ∂M \ ( R + ∪ R − ) is acollection of annuli and tori where these annuli join ∂R + to ∂R − . These complementary annuli and tori arecalled the sutures . The orientation on R + is taken consistent with the boundary orientation of ∂M while R − is oriented oppositely. We will further assume throughout that no component of R + or R − is a sphere. Definition 2.2 (Compression bodies) . Following [CG87], a compression body W is a cobordism rel- ∂ betweenorientable surfaces ∂ + W and ∂ − W that may be formed as follows: there is a non-empty compact orientablesurface S (without sphere components) such that W is assembled from the product S × [ − ,
1] by attaching2–handles to S × {− } and then 3–handles to any resulting sphere boundary components meeting S × {− } that are disjoint from ∂S × [ − , ∂ v W = ∂S × [ − ,
1] is the vertical boundary of W , and the complement in ∂W of its interior are the surfaces ∂ + W = S × { +1 } and ∂ − W . Note that(1) a compression body need not be connected,(2) ∂ + W is necessarily non-empty, and(3) ∂ − W has no sphere components.Dually, we may view a compression body W as formed as follows: there is an orientable surface F (without sphere components, and possibly F = ∅ ) such that W is assembled from the disjoint union ofthe product F × [ − , +1] and a collection of 0–handles by attaching a collection of 1–handles to F × { +1 } and the boundaries of these 0–handles. (Furthermore, every 0–handle has a 1–handle attached to it.) Here ∂ v W = ∂F × [ − , ∂W of its interior are the surfaces ∂ − W = F × {− } , and ∂ + W .If W is connected, then the components of F × [ − , +1] and the 0–handles are all joined by a sequence of1–handles. Duality exchanges the 0– and 3–handles and the 1– and 2–handles.A trivial compression body W is a product W = S × [ − ,
1] so that ∂ + W = S × { } . A connectedcompression body W with ∂ − W = ∅ is a handlebody . Remark 2.3.
For comparison, Bonahon introduced compression bodies W for which ∂ + W is a closed andpossibly disconnected surface. Furthermore he allows ∂ + W to have a sphere component as long as it boundsa ball. See [Bon83]. Remark 2.4.
A compression body W may be regarded as a special kind of sutured manifold where R + = ∂ + W and R − = ∂ − W , and the annuli ∂ v W are the sutures. Furthermore, if W is a connected compressionbody with ∂W connected, then W is also a handlebody if we forget the sutured manifold structure (that is,if we set ∂ + W = ∂W ). Throughout, our compression bodies will retain their sutured manifold structure. — 2.2 Heegaard splittings —Definition 2.5 (Heegaard splittings) . Let M be a connected sutured manifold without toroidal sutures.A Heegaard splitting of M , denoted ( M, S ), is a decomposition of M along an oriented properly embeddedsurface S into a pair of non-empty connected compression bodies A and B so that M = A ∪ B where S = A ∩ B = ∂ + A = ∂ + B , R − = ∂ − A , and R + = ∂ − B . (Here we flip the orientation of B as a suturedmanifold.) The surface S = A ∩ B is the Heegaard surface of the splitting, and this surface defines thesplitting. For notational convenience we also refer to the Heegaard splitting as the triple (
A, B ; S ). Observe hat the boundary components of S may be regarded as the core curves of the annular sutures of M . Furthernote that S is necessarily connected.A Heegaard splitting ( A, B ; S ) is trivial if either A or B is a trivial compression body.Any connected sutured manifold M without toroidal sutures has a Heegaard splitting as long as neither R + nor R − has a sphere component, cf. [CG87, God92]. Definition 2.6 (Circular Heegaard splitting) . Let M be a connected sutured manifold, possibly with toroidalsutures. A circular Heegaard splitting of M , denoted ( M, R, S ), is a decomposition along a pair of disjoint,properly embedded, oriented surfaces R and S so that: • ∂R is contained in the sutures of M , • M \ R is a connected sutured manifold without toroidal sutures, and • S is a Heegaard surface for M \ R . Definition 2.7 (Generalized Heegaard splittings) . Let M be a sutured manifold, possibly with toroidalsutures. A generalized Heegaard splitting of M , denoted ( M, R , S ), is a decomposition of M along a pair ofdisjoint, properly embedded, oriented surfaces R and S satisfying the following conditions: • ∂ R is in the sutures of M ; • R decomposes M into a collection M \R of connected sutured submanifolds without toroidal sutures; • S intersects each component of M \R in a single Heegaard surface; and • the compression bodies of M \ ( R ∪ S ) may be labeled A or B so that each component of R ∪ S meetsboth a A compression body and a B compression body.The surfaces R and S are commonly called “thin” and “thick” respectively, as are their components. Notethat each component of R is both a component of R + ( M i ) and a component of R − ( M j ) for some components M i , M j of M \R (allowing M i = M j ). We may choose the labeling of the compression bodies of M \ ( R ∪ S )so that for any component M i of M \R , the A compression body of its splitting by S meets R − ( M i ) and the B compression body meets R + ( M i ).Observe that if R = ∅ , then the generalized Heegaard splitting ( M, ∅ , S ) is just a Heegaard splitting of M . Indeed, when R = ∅ then M \R = M so that M necessarily has no toroidal sutures and S is just aHeegaard surface for M .The adjective generalized indicates that R may split M into possibly more than one connected suturedsubmanifold. Remark 2.8.
Often in the literature, the term “generalized Heegaard splitting” allows for disconnectedcompression bodies in Heegaard splittings of potentially disconnected manifolds. In these contexts, typicallythe thin and thick surfaces R and S are partitioned as regular levels of a Morse function of M to either R or S appearing alternately. Between a consecutive pair of regular thin levels of R is a (possibly disconnected)manifold for which the intermediate regular thick level of S restricts to a Heegaard surface on each component.This Morse-theoretic version of generalized Heegaard splittings may be recovered from our kind of generalizedHeegaard splitting by inserting pairs of trivial compression bodies as needed and taking their unions. Onemay add the adjective linear or circular to the term “generalized Heegaard splitting” to emphasize that thesplitting is associated to a Morse function to R or S respectively. Note that it is possible for the samegeneralized Heegaard splitting to be regarded as both linear and circular.Our approach to generalized Heegaard splittings falls more in line with that of [SSS16, Definition 4.1.7],though we do not require exactness of the associated fork complex (see [SSS16, Definitions 4.1.3 & 4.1.4]).Furthermore, we allow our compression bodies to have vertical boundary. — 2.3 Weak Reductions and Amalgamations — Explicit descriptions of weak reductions andamalgamations are given in [Sch93, §
2] in the context of compression bodies without vertical boundary. Wepresent them here slightly differently while allowing vertical boundary, but the idea is basically the same.
Definition 2.9 (Stabilized splittings) . Let M be a connected sutured manifold without toroidal sutures.Then both the Heegaard splitting ( A, B ; S ) and the Heegaard surface S are called stabilized if S has com-pressing disks D A in A and D B in B such that ∂D A and ∂D B are transverse in S and intersect exactly once.In such a situation, a compression of S along D A (or D B ) produces a new Heegaard surface S (cid:48) of genus oneless (allowing for the splitting of S into two balls). The reduction from S to S (cid:48) is called a destabilization .The inverse process (of tubing a Heegaard surface along a boundary parallel arc) is called a stabilization . efinition 2.10 (Reducible and irreducible splittings) . Let M be a connected sutured manifold withouttoroidal sutures. Then both the Heegaard splitting ( A, B ; S ) and the Heegaard surface S are called reducible if S has compressing disks D A in A and D B in B such that ∂D A = ∂D B . They are irreducible if they arenot reducible.Note that a stabilized splitting is reducible. By Waldhausen [Wal68], a reducible splitting of an irreduciblemanifold is stabilized. Definition 2.11 (Weakly reducible & strongly irreducible) . Let M be a connected sutured manifold withouttoroidal sutures. Then both the Heegaard splitting ( A, B ; S ) and the Heegaard surface S are called weaklyreducible if S has compressing disks D A in A and D B in B that are disjoint. They are called stronglyirreducible otherwise. That is, a Heegaard splitting ( A, B ; S ) is strongly irreducible if every compressing diskof S in A intersects every compressing disk of S .If some component of S of a generalized Heegaard splitting ( M, R , S ) is a weakly reducible Heegaardsurface for its component of M \R , then the generalized Heegaard splitting is weakly reducible . Otherwise,the generalized Heegaard splitting is strongly irreducible . (In [SS00, Definition 2.9], it is further required thata strongly irreducible generalized Heegaard splitting has R incompressible. We call such a splitting locallythin, see Definition 2.15.) Remark 2.12.
Vacuously, a trivial Heegaard splitting is strongly irreducible by definition. In contrast,where the term was first introduced at the beginning of [CG87, § Lemma 2.13.
Let M be a connected sutured manifold without toroidal sutures. If R + ∪ R − is compressible,then any non-trivial Heegaard splitting of M is weakly reducible. Consequently, if ( M, R , S ) is a generalizedHeegaard splitting in which R ∪ R + ∪ R − is compressible, then the splitting is weakly reducible.Proof. The result follows from [CG87, Theorem 2.1] and Definition 2.11. (cid:3)
Remark 2.14.
Any weakly reducible Heegaard splitting may easily be “inflated” to a strongly irreduciblegeneralized Heegaard splitting. Let S be a weakly reducible Heegaard surface for M , and consider a closedproduct neighborhood S × [ − ,
1] of S in M . Setting R = S × { } and S = S × { +1 } ∪ S × {− } , thegeneralized Heegaard splitting ( M, R , S ) is strongly irreducible as the components of S give trivial splittingsof the two compression bodies M \R . Note that, however, R is compressible. (Such an “inflation” may beregarded as a trivial weak reduction, see Definition 2.16.) Definition 2.15 (Locally thin, cf. [MG13, §
2] and [EMnMG12, Definition 3.2]) . A generalized Heegaardsplitting ( M, R , S ) is locally thin if it is strongly irreducible and R is incompressible. Definition 2.16 (Weak reduction) . Let ( M, R , S ) be a weakly reducible but irreducible generalized Heegaardsplitting of an irreducible sutured manifold M . Assume the component S of S defines an irreducible butweakly reducible Heegaard splitting ( A, B ; S ) of its component M (cid:48) of M \R . A weak reduction of ( M, R , S )along S is a generalized Heegaard splitting ( M, R (cid:48) , S (cid:48) ) obtained as follows.Let D A and D B be collections of disjoint compressing disks for S in A and B respectively such that(1) ∂D A and ∂D B are disjoint in S , and(2) no component of S \ ( ∂D A ∪ ∂D B ) is a planar surface P whose boundary is contained in either only ∂D A or only ∂D B (possibly with duplicity).Consider a regular neighborhood N = N ( S ∪ D A ∪ D B ) of S and the disks D A and D B and the twosubmanifolds of M (cid:48) to either side of N , A = ( M (cid:48) \ N ) ∩ A and B = ( M (cid:48) \ N ) ∩ B . Within N is the result R of compressing S along D A and D B . Then this surface R splits N into two (possibly non-connected)compression bodies B and A which meet A and B along surfaces S and S respectively. Consequently R splits M (cid:48) into two (possibly non-connected) sutured submanifolds M (cid:48) and M (cid:48) for which ( A i , B i ; S i ) restrictsto a Heegaard splitting on each component of M (cid:48) i where i = 1 ,
2. This gives the generalized Heegaardsplitting ( M, R (cid:48) , S (cid:48) ) in which R (cid:48) = R ∪ R and S (cid:48) = ( S − S ) ∪ ( S ∪ S ).By the proof of [CG87, Theorem 3.1] (see also its treatment in the proof of [Sch03, Theorem 3.11]), ifthe Heegaard surface S is irreducible, then the disk sets D A and D B may be chosen so that the new surface resulting from the weak reduction is incompressible in M . In such a case we say the weak reduction is maximal . That is, ( M, R (cid:48) , S (cid:48) ) results from a maximal weak reduction of ( M, S , R ) along S . Remark 2.17.
Observe that in Definition 2.16, the new component R of R (cid:48) resulting from a maximal weakreduction along a weakly reducible but irreducible surface S need not be connected. Moreover, it is possiblethat one of the new components S or S of S (cid:48) produced by the maximal weak reduction is itself weaklyreducible (cf. the paragraph after the proof of [Sch03, Theorem 3.11]), though it is necessarily irreducible. Remark 2.18.
In lieu of condition (2) in our Definition 2.16, some authors simply discard any spherecomponent that arises when compressing S by D A ∪ D B . By our assumption of irreducibility of the manifold M , such sphere components either imply that S is reducible or a disk of D A or D B is superfluous. Definition 2.19 (Iterated weak reduction) . If a generalized Heegaard splitting ( M, R , S ) is weakly reduciblebut irreducible, then some component of S is weakly reducible for its component of M \R . Hence a weakreduction may be performed for that component. Consequently we may have a sequence of weak reductions( M, R , S ) = ( M, R , S ) (cid:55)→ ( M, R , S ) (cid:55)→ · · · (cid:55)→ ( M, R n , S n )where each weak reduction ( M, R i , S i ) (cid:55)→ ( M, R i +1 , S i +1 ) is performed along a weakly reducible componentof S i . In particular, the weak reduction performed along the weakly reducible component S of S i producesthree surfaces S , R , and S so that R i +1 = R i ∪ R and S i +1 = ( S i − S ) ∪ ( S ∪ S ). The resulting generalizedHeegaard splitting ( M, R n , S n ) is an iterated weak reduction of ( M, R , S ). If no component of S n is weaklyreducible and each weak reduction of the sequence is maximal, then ( M, R n , S n ) is a maximal iterated weakreduction of ( M, R , S ). Lemma 2.20.
Let M be a sutured manifold with irreducible Heegaard surface S . Let ( M, R , S ) be a maximaliterated weak reduction of ( M, ∅ , S ) . Then ( M, R , S ) is a locally thin generalized Heegaard splitting.Proof. Since ( M, R , S ) is the result of a maximal iterated weak reduction, the components of S are allstrongly irreducible by Definition 2.19. Hence the generalized Heegaard splitting is strongly irreducible.Furthermore, as noted in Remark 2.17, the components of R produced by the maximal weak reductions areall incompressible. Hence the generalized splitting is locally thin. (cid:3) Definition 2.21 (Amalgamation) . The inverse process of a weak reduction is called an amalgamation .More specifically, we may speak of amalgamations of a generalized Heegaard splitting ( M, R , S ) along certainunions of components of R . In addition to [Sch93, § §
3] for another treatment of amalgamation.Let ( M, R , S ) be a generalized Heegaard splitting. For i = 1 ,
2, let M i be two distinct connected com-ponents of M \R with Heegaard splittings ( A i , B i ; S i ) where S i is a component of S . Assume the surface R = B ∩ A ⊂ R is non-empty. ( R may be disconnected.) We first define the amalgamation along R of thegeneralized Heegaard splitting ( M ∪ R M , R, S ∪ S ) to be the Heegaard splitting ( M ∪ R M , S (cid:48) ) obtainedas follows:View the compression bodies A and B as being assembled from attaching 1–handles to either side ofa thickening R × [ − ,
1] of R . Note that we may assume that the feet of these 1–handles have mutuallydisjoint projections to R = R × { } ⊂ R × [ − , R × [ − ,
1] so that their feet now meet the other side. In particular, this means that thefeet of the 1–handles of A are in S and the feet of the 1–handles of B are in S . Observe that S tubed along these 1–handles of A is a connected surface that is also isotopic to S tubed along these1–handles of B . In fact, extending the feet of the one-handles of both collections to only R = R × { } and then tubing R along these 1–handles produces a surface S (cid:48) that is also isotopic to them.We now see that S (cid:48) is a Heegaard surface for M ∪ R M . View the compression body A as alsobeing assembled from 1–handles attached to ∂ − A and a collection of 0–handles. Then the feet of the1–handles from A (now in S ) can be slid off the 1–handles of A down to the surface ∂ − A and the0–handles. Thus we have a new handlebody A (cid:48) with ∂ − A (cid:48) = ∂ − A and ∂ + A (cid:48) = S (cid:48) . Similarly we forma new handlebody B (cid:48) with ∂ − B (cid:48) = ∂ − B and ∂ + B (cid:48) = S (cid:48) by viewing B as assembled from 1–handlesand sliding the 1–handles from B off them. Hence we obtain the Heegaard splitting ( M ∪ R M , S (cid:48) ).Now let R (cid:48) = R − R and S (cid:48) = S ∪ S (cid:48) − ( S ∪ S ). Then we define the amalgamation along R of ( M, R , S )to be the generalized Heegaard splitting ( M, R (cid:48) , S (cid:48) ). inally, an amalgamation of a generalized Heegaard splitting ( M, R , S ) is the result of a sequence ofamalgamations along surfaces as above. Remark 2.22.
Continuing with the notation above, we may also consider amalgamations along R when R is only a proper subset of components of B ∩ A . In this situation we may first “inflate” the generalizedHeegaard splitting ( M, R , S ) by inserting a product manifold with the trivial splitting for each component of( B ∩ A ) − R so that B and A meet only along R . Then we take the amalgamation along R of ( M, R , S )to be the amalgamation along R of the “inflated” generalized Heegaard splitting. While we will not needthis extension of amalgamation, it is interesting to note that it changes the homotopy class of the circularMorse function associated to the generalized Heegaard splitting. — 2.4 Handle numbers and handle indices —Definition 2.23 (Handle number) . Following Goda [God06, Definition 3.4], the handle number of a com-pression body W is the minimum number h ( W ) of 1–handles needed in its construction. In particular, if W is connected, then h ( W ) = g ( ∂ + W ) − g ( ∂ − W ) + | ∂ − W − | where g ( ∂ − W ) is the sum of the genera of the components of ∂ − W and ∂ − W is the number of its compo-nents. (Set g ( ∅ ) = 0.) Viewing W as built from attaching 1–handles to ∂ − W and 0–handles, one observes • exactly one 0–handle is needed if ∂ − W = ∅ (so that W is a handlebody) and • no 0–handles are needed if ∂ − W (cid:54) = ∅ .If W is disconnected, then h ( W ) is the sum of the handle numbers of its components.Let M be a sutured manifold. The handle number h ( M, S ) of a Heegaard splitting (
M, S ) is the sumof the handle numbers of the compression bodies of M \ S . The handle number h ( M, R , S ) of a generalizedHeegaard splitting ( M, R , S ) is the sum of the handle numbers of the compression bodies of M \ ( R ∪ S ).The handle number h ( R ) of a Seifert surface R for an oriented link is the minimum of h ( M, R, S ) = h ( M \ R, S ) among circular Heegaard splittings (
M, R, S ) of the link exterior M . The handle number h ( L )of an oriented link L is the minimum of h ( R ) among Seifert surfaces of L Attention:
Our handle numbers h ( R ) and h ( L ) are twice what Goda defines. Proposition 2.24 ( [God06, Proposition 3.7]) . If L is an oriented link in S , then MN ( L ) = h ( L ) . (cid:3) Definition 2.25 (Handle index) . The handle index j ( W ) of a compression body W built by attaching1–handles to ∂ − W and a collection of 0–handles is the number j ( W ) = − . Dually, if W is built by attaching 2–handles to ∂ + W and filling a collection of resulting 2–spheres with3–handles, then j ( W ) = − . Equivalently, the handle index j ( W ) of a compression body W is its handle number h ( W ) minus the numberof handlebodies in W .Let M be a sutured manifold. The handle index j ( M, S ) of a Heegaard splitting (
M, S ) is the sum of thehandle indices of the compression bodies of M \ S . The handle index j ( M, S , R ) of a generalized Heegaardsplitting ( M, S , R ) is the sum of the handle indices of the compression bodies of M \ ( R ∪ S ). Remark 2.26.
In [SS01, Definition 2.1] and [SS00, Definition 2.5], Scharlemann and Schultens define thehandle index of a (connected) compression body to be J ( W ) = χ ( ∂ − W ) − χ ( ∂ + W ). This index is twice ourhandle index. That is, J ( W ) = 2 j ( W ). Lemma 2.27.
For generalized Heegaard splittings, the handle index is preserved by weak reductions andamalgamations.Proof.
While weak reductions and amalgamations may create or cancel of 0– and 1–handle pairs or 2– and 3–handle pairs, these operations never create or cancel 1– and 2–handle pairs. Hence these operations preservethe handle index. (cid:3) igure 1. Shown is a schematic of the sequence of a maximal weak reduction followed byamalgamations that transforms a circular Heegaard splitting (
M, R, S ) with R compressibleinto a circular Heegaard splitting ( M, R , S (cid:48)(cid:48) ) with R incompressible and the same handlenumber. The central loops/arcs (shown in red) represent the toroidal/annular sutures andthe vertical boundary of the compression bodies. Lemma 2.28.
Any two generalized Heegaard splittings without handlebodies that are related by a sequenceof weak reductions and amalgamations have the same handle number.Proof.
Since h ( W ) = j ( W ) for a connected compression body W that is not a handlebody, this is animmediate consequence of Lemma 2.27 (cid:3) First we show that the handle number of a knot may be realized by an incompressible Seifert surface.
Proof of Lemma 1.2.
By Definition 2.23, K has a Seifert surface R such that h ( K ) = h ( R ). If R is incom-pressible, then we are done. So assume R is compressible. Figure 1 provides a schematic of the remainderof the proof in this case.Let ( M, R, S ) be an associated circular Heegaard splitting of the knot exterior M = S \N ( K ) real-izing h ( R ). That is, h ( M, R, S ) = h ( R ). Since R is compressible, this splitting is weakly reducible byLemma 2.13. Therefore a maximal weak reduction of S produces a generalized circular Heegaard splitting( M, R , S ) = ( M, R ∪ R , S ∪ S ) where R = R and R is an incompressible and possibly disconnected sur-face. Furthermore, we may view R as dividing the connected sutured manifold M \ R into two (potentiallydisconnected) sutured manifolds M and M with with Heegaard splittings ( A , B ; S ) and ( A , B ; S )respectively such that R = B ∩ A and R = B ∩ A . (More specifically, for each i = 1 ,
2, on eachcomponent of M i the surface S i restricts to a connected Heegaard surface dividing that component of M i into a compression body of A i and a compression body of B i .)Since S is a Seifert surface for K , it has a single boundary component. Because the incompressible surface R is obtained by compressing S , it also has a single boundary component. Hence exactly one componentof R , say R , has boundary and any other components are closed. In particular, R is an incompressibleSeifert surface for K . We will show that h ( K ) = h ( R ). ndex the closed components of R , if any, as R i with integers i = 1 , , . . . , k as needed. Observe thateach of these closed components R i for i ≥ M . Hence each closed component R i of R is the boundary of a submanifold of either of the following forms:(1) A i ∪ S i B i where A i is a handlebody component of A and B i is a connected component of thecompression body B , or(2) A i ∪ S i B i where B i is a handlebody component of B and A i is a connected component of thecompression body A .Perform amalgamations along these closed components of R . This produces a circular generalized Hee-gaard splitting ( M, R (cid:48) , S (cid:48) ) = ( M, R (cid:48) ∪ R (cid:48) , S (cid:48) ∪ S (cid:48) ) in which the components R (cid:48) , S (cid:48) , R (cid:48) , and S (cid:48) are eachnon-separating. Indeed, the amalgamations merge the surfaces S − S with S to produce S (cid:48) and S − S with S to produce S (cid:48) , leaving R = R (cid:48) and R = R (cid:48) unaffected. (Since an amalgamation is the inverseprocess of a weak reduction, one may obtain this splitting ( M, R (cid:48) , S (cid:48) ) as a weak reduction of the originalsplitting ( M, R, S ).)Now amalgamate ( M, R (cid:48) , S (cid:48) ) along R to obtain a splitting ( M, R , S (cid:48)(cid:48) ). Since the surface S (cid:48)(cid:48) is anamalgamation of two connected surfaces, it too is connected. Since the ( M, R , S (cid:48)(cid:48) ) and ( M, R, S ) aresplittings without handlebodies that are related by a sequence of weak reductions and amalgamations, theyhave the same handle number by Lemma 2.28. Hence h ( R ) = h ( M, R , S (cid:48)(cid:48) ) = h ( M, R, S ) = h ( R ). Therefore R is an incompressible Seifert surface for K such that h ( K ) = h ( R ). (cid:3) Next we prove the additivity of Morse-Novikov number for knots by showing their handle number isadditive. For this we recall key results of Eudave-Munoz and Manjarrez-Gutierrez about locally thin circulargeneralized Heegaard splittings of exteriors of connected sums of knots.
Theorem 3.1 (Proposition 5.1 and Corollary 5.3 [EMnMG12]) . Let M be the exterior of K a K b . Let Q ⊂ M be the properly embedded annulus that decomposes M into the exteriors M a and M b of K a and K b .If ( M, R , S ) is a locally thin circular generalized Heegaard splitting, then R ∪ S can be isotoped in M so that(1) Q \ ( R ∩ S ) is a collection of vertical rectangles in the compression bodies of the splitting, and(2) the restriction ( M i , R ∩ M i , S ∩ M i ) is a circular generalized Heegaard splitting for each i = a, b . (cid:3) Proof of Theorem 1.1.
Using Goda’s interpretation of the Morse-Novikov number in terms of handle number(see Proposition 2.24), we must show that h ( K a K b ) = h ( K a ) + h ( K b ). Let M = S \N ( K ) be the exteriorof K , and let M i = S \N ( K i ) be the exterior of K i for each i = a, b .The inequality h ( K a K b ) ≤ h ( K a )+ h ( K b ) is straightforward as implied in [Paj06, Section 14.6.2]. Indeedfor each i = a, b let R i be a Seifert surface for K i such that h ( K i ) = h ( R i ). Then each knot exterior M i has acircular Heegaard splitting ( M i , R i , S i ) realizing its handle number. The knot exterior M may be obtained bygluing M a to M b along a closed regular annular neighborhood of a meridian in each of their boundaries. Thisgluing may be done so that ∂R a and ∂R b meet along a single arc to form a boundary sum R = R a (cid:92)R b that isa Seifert surface for K . Similarly ∂S a and ∂S b meet to form a boundary sum S = S a (cid:92)S b . Together they give acircular Heegaard splitting ( M, R, S ). From this we find that h ( K ) ≤ h ( R ) = h ( R a )+ h ( R b ) = h ( K a )+ h ( K b ).For the other inequality, by Lemma 1.2 we may assume there is an incompressible Seifert surface R forthe knot K = K a K b such that h ( K ) = h ( R ). Then there is an associated circular Heegaard splitting( M, R, S ) realizing this handle number. Because S is connected, the two compression bodies of this splittingare connected and neither is a handlebody. Let ( M, R , S ) be a locally thin generalized circular Heegaardsplitting resulting from a maximal iterated weak reduction as guaranteed by Lemma 2.20. Then h ( K ) = h ( R ) = h ( M, R, S ) = j ( M, R, S ) = j ( M, R , S )where the first two equalities are by definition, the third follows from Definition 2.25 because ( M, R, S ) hasno handlebodies in its splitting, and the fourth is due to Lemma 2.27.Let Q be the summing annulus that splits M into the exteriors of K a and K b . That is, Q is the annulusalong which M a and M b are identified to form M ; M \ Q = M a (cid:116) M b . Since the splitting ( M, R , S ) is locallythin, Theorem 3.1 shows that we may arrange that Q chops the splitting into circular generalized Heegaardsplittings for M a and M b . In particular the restriction of ( M, R , S ) to M i for i = a, b is the splitting M i , R i , S i ) where R i = R ∩ M i and S i = S ∩ M i . Observe that a (not necessarily connected) compressionbody obtained by chopping another compression body along a vertical rectangle has the same handle index.Hence j ( M, R , S ) = j ( M a , R a , S a ) + j ( M b , R b , S b ) . For each i = a, b , a sequence of amalgamations brings ( M i , R i , S i ) to a circular Heegaard splitting ( M i , R i , S i )with no handlebodies so that j ( M i , R i , S i ) = j ( M i , R i , S i ) = h ( M i , R i , S i ) ≥ h ( R i ) ≥ h ( K i ) . (Note that if ( M i , R i , S i ) had a handlebody in its decomposition, then S i would have a closed componentbounding this handlebody. Then the component of the compression body meeting the other side of S i wouldhave a closed component of R i in its negative boundary. But then an amalgamation could be performedalong this closed component.) Thus we have h ( K ) ≥ h ( K a ) + h ( K b )as desired. (cid:3) Definition 4.1 (Patterns and satellites) . An oriented two-component link k ∪ c in S where c is an unknot,defines a pattern P which is the knot k in the solid torus S \N ( c ) equipped with a longitude that is themeridian of c . Then, given a knot K and a pattern P , the satellite knot P ( K ) is obtained by replacing aregular solid torus neighborhood of K with P so that the longitudes agree.Let P be a pattern defined by the link k ∪ c . The winding number of P is the linking number of k with c . We say P is a fibered pattern if k is a fibered knot and c is transverse to the fibration. Definition 4.2 (Circular Heegaard splitting and handle number of a pattern) . If the winding number of P is non-zero, then we may define a circular Heegaard splitting of P to be a circular Heegaard splitting( M, R, S ) of the link exterior M = S \N ( k ∪ c ) viewed as a sutured manifold with two toroidal suturesin which R = R \N ( c ) for a Seifert surface R of k . In particular, ( M, R, S ) is the restriction of a circularHeegaard splitting (
M , R, S ) of the knot exterior M = S \N ( k ) where c intersects the two compressionbodies of M \ ( R ∪ S ) in unions of vertical arcs.Then we may take the handle number h ( P ) of P to be the minimum handle number among circularHeegaard splittings of P .The following is an immediate consequence of definitions. Lemma 4.3.
For a pattern P with non-zero winding number, h ( P ) = 0 if and only if P is fibered. (cid:3) Lemma 4.4.
Let P be a pattern with non-zero winding number, and let K be a knot in S . Then h ( P ( K )) ≤ h ( P ) + h ( K ) .Proof. We may build a generalized Heegaard splitting for P ( K ) by gluing together circular Heegaard splittingfor P with an inflated circular Heegaard splitting for K . Then the handle number of the assembled generalizedHeegaard splitting for P ( K ) will be the sum of the handle numbers of the two circular Heegaard splittings.The result will then follow by using circular Heegaard splittings that realize the handle numbers of P and K .Assume P is defined by the link k ∪ c where c is the unknot and the winding number is n >
0. Then theexterior M = S \N ( P ( K )) of P ( K ) is the union of M P = S \N ( k ∪ c ) and M K = S \N ( K ) along the tori ∂ N ( c ) and ∂ N ( K ) so that the meridian of c is identified with the longitude of K while the longitude of c isidentified with the reversed meridian of K .Let ( M P , R P , S P ) be a circular Heegaard splitting for P . Then the thin and thick surfaces R P and S P chop M P into two connected compression bodies A P and B P . Each A P and B P has n + 1 vertical boundarycomponents; one is a longitudinal annulus of ∂ N ( k ) while the other n are meridional annuli of ∂ N ( c ).Let ( M K , R K , S K ) be a circular Heegaard splitting for K . Then the thin and thick surfaces R K and S K chop M K into two connected compression bodies A K and B K whose vertical boundaries are each asingle longitudinal annulus of ∂ N ( K ). Inflate this splitting n − M K , R K , S K ) where R K consists of n parallel copies of R K and S K consists of S K and n − arallel copies of R K alternately between those of R K . This inserts 2( n −
1) trivial compression bodies(homeomorphic to R K × [ − , R K .Now when we join M K to M P to form the exterior of P ( K ), we may glue ( M K , R K , S K ) to ( M P , R P , S P )so that ∂ R K identifies with ∂R P ∩ ∂ N ( c ) and ∂ S K identifies with ∂S P ∩ ∂ N ( c ). This produces Seifertsurfaces for P ( K ) where R is R P with n copies of R K attached along ∂R P ∩ ∂ N ( c ) and S is S P with S K and n − R K attached along ∂S P ∩ ∂ N ( c ). Furthermore, this then causes the compression bodies A K and B K and the 2( n −
1) trivial compression bodies of M K \ ( R K ∪ S K ) to be attached to A P and B P alongtheir vertical boundaries that are the meridional annuli of ∂ N ( c ). Since gluing two compression bodies W and W together along a vertical boundary component (so that ∂∂ + W meets ∂∂ + W ) produces anothercompression body, ( M, R, S ) is a circular Heegaard splitting for M , the exterior of P ( K ).By construction, h ( M, R, S ) = h ( M P , R P , S P ) + h ( M K , R K , S K ). Hence, choosing to use circular Hee-gaard splittings for which h ( M P , R P , S P ) = h ( P ) and h ( M K , R K , S K ) = h ( K ), then we obtain h ( P ( K )) ≤ h ( P ) + h ( K ). (cid:3) Definition 5.1 (Cables) . For coprime integers p, q with p >
0, let k be a ( p, q )–torus knot in the Heegaardtorus T of S . One of the core curves of the solid tori bounded by T has linking number p with k , and we let c be that unknot. Then the link k ∪ c defines the ( p, q ) –cable pattern P p,q . For a knot K , its ( p, q ) –cable isthe satellite knot K p,q = P p,q ( K ). The cabling annulus for the cabled knot K p,q is the image of the annulus T − N ( k ) of P p,q − N ( k ) in the exterior S \N ( K p,q ) of the cabled knot. Lemma 5.2. P p,q is a fibered pattern.Proof. This is rather well-known. Let k ∪ c be the link described in Definition 5.1. Then there is a Seifertfibration of S in which k is a regular fiber and c is an exceptional fiber of order p . (If p = 1 then c mayalso be regarded as a regular fiber.) This Seifert fibration restricts to a Seifert fibration on the exterior of k .Since torus knots are fibered knots, the exterior of k also fibers as a surface bundle over S . As the Seifertfibration is transverse to this fibration, we see that c is transverse to the fibration. Hence P p,q is a fiberedpattern. (cid:3) The proof of Theorem 3.1 given in [EMnMG12] extends directly for any properly embedded essentialannulus in a manifold with locally thin circular generalized Heegaard splitting as long as no boundary curveof the annulus is isotopic to a boundary curve of the thin surface. We record this extension here as we willapply it with a cabling annulus.
Theorem 5.3 (Cf. Theorem 3.1 and [EMnMG12, Proposition 5.1 and Corollary 5.3]) . Let M be an irreduciblesutured manifold with toroidal sutures. Let Q be a properly embedded essential annulus with ∂Q in the toroidalsutures of M . If ( M, R , S ) is a locally thin circular generalized Heegaard splitting such that no curve of ∂Q is isotopic in ∂M to a curve of ∂ R , then R ∪ S can be isotoped in M so that • Q ∩ ( R ∪ S ) is a collection of vertical rectangles in the compression bodies of the splitting, and • chopping along Q renders ( M, R , S ) into a circular generalized Heegaard splitting ( M \ Q, R\ Q, S\ Q ) . (cid:3) Note that, though we do not need it here, we have stated this theorem to allow for Q to be non-separatingand even for ∂Q to be contained in different components of ∂M . Proof of Theorem 1.3.
Recall p, q are coprime integers with p >
0. We aim to show that h ( K p,q ) = h ( K ) fora knot K . By Goda (Proposition 2.24), the corresponding statement for Morse-Novikov number will holdtoo.First off, if p = 1 then K p,q = K , and so the result is trivial. Hence we assume p ≥ p, q )–cable is a fibered pattern by Lemma 5.2, it follows from Lemmas 4.3 and4.4 that h ( K p,q ) ≤ h ( K ). Hence we must show h ( K p,q ) ≥ h ( K ).Let M = S \N ( K p,q ) be the exterior of the cabled knot K p,q , let M K = S \N ( K ), and let M V be a solidtorus (the exterior of the unknot). The cabling annulus Q is a properly embedded annulus that chops M into M K and M V ; M \ Q = M K (cid:116) M V . The proof now follows similarly to that of Theorem 1.1. et ( M, R, S ) be a circular Heegaard splitting for the exterior of K p,q such that h ( K p,q ) = h ( M, R, S ).By iterated maximal weak reductions we may obtain a locally thin generalized Heegaard splitting ( M, R , S )realizing this handle number as guaranteed by Lemma 2.20. Then h ( K p,q ) = h ( M, R, S ) = j ( M, R, S ) = j ( M, R , S ) . Since p ≥
2, the boundary components of the cabling annulus Q have slope distinct from the boundaryslope of a Seifert surface . We may now apply Theorem 5.3 to obtain the generalized Heegaard splitting( M \ Q, R\ Q, S\ Q ). As in the proof of Theorem 1.1, it follows that j ( M, R , S ) = j ( M \ Q, R\ Q, S\ Q ) . Because M \ Q = M K (cid:116) M V , we may divide the generalized Heegaard splitting ( M \ Q, R\ Q, S\ Q ) intogeneralized Heegaard splittings ( M K , R K , S K ) and ( M V , R V , S V ). Hence j ( M, R , S ) = j ( M K , R K , S K ) + j ( M V , R V , S V ) . For each i = K, V , a sequence of amalgamations along closed components of R i brings ( M i , R i , S i ) to acircular Heegaard splitting ( M i , R i , S i ) with no handlebodies so that j ( M i , R i , S i ) = j ( M i , R i , S i ) = h ( M i , R i , S i ) . Then h ( M K , R K , S K ) ≥ h ( K ) and h ( M V , R V , S V ) ≥
0. Hence h ( K p,q ) ≥ h ( K ) + 0 = h ( K )as desired. (cid:3) Remark 5.4.
Alternatively, at the end of the above proof, instead of chopping along Q once it is positioned tointersect the compression bodies of the generalized Heegaard splitting ( M, R , S ) in only vertical rectangles,one may prefer to first amalgamate to a circular Heegaard splitting ( M, R (cid:48) , S (cid:48) ) while preserving the nicestructure of how Q intersects the splittings. Nevertheless, chopping this circular Heegaard splitting along Q necessarily produces two generalized Heegaard splittings, and (unless p = 1) amalgamations would still beneeded to obtain circular Heegaard splittings.Indeed, when a Seifert surface for K p,q (such as an incompressible Seifert surface) may be isotoped tointersect Q only in spanning arcs, it does so in | pq | arcs. Then Q chops the surface into p Seifert surfacesfor K and | q | Seifert surfaces for the unknot.
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Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA
E-mail address : [email protected]@math.miami.edu