On Floer homology and the Berge conjecture on knots admitting lens space surgeries
aa r X i v : . [ m a t h . G T ] O c t ON FLOER HOMOLOGY AND THE BERGE CONJECTURE ONKNOTS ADMITTING LENS SPACE SURGERIES
MATTHEW HEDDEN
Abstract.
We complete the first step in a two-part program proposed by Baker,Grigsby, and the author to prove that Berge’s construction of knots in the three-spherewhich admit lens space surgeries is complete. The first step, which we prove here, isto show that a knot in a lens space with a three-sphere surgery has simple (in thesense of rank) knot Floer homology. The second (conjectured) step involves showingthat, for a fixed lens space, the only knots with simple Floer homology belong to asimple finite family. Using results of Baker, we provide evidence for the conjecturalpart of the program by showing that it holds for a certain family of knots. Coupledwith work of Ni, these knots provide the first infinite family of non-trivial knots whichare characterized by their knot Floer homology. As another application, we provide aFloer homology proof of a theorem of Berge. Introduction
On which knots in the three-sphere can one perform Dehn surgery and obtain a lensspace? This question has received considerable attention in recent years [4, 5, 19, 24, 10,30, 1, 2] and much progress has been made towards a general method of enumerationof such knots. Indeed, there is a conjecture that a construction due to Berge [4] whichproduces knots in S with lens space surgeries is complete (in the sense that any knotadmitting a lens space surgery comes from this construction). The purpose of this paperis discuss a strategy by which the knot Floer homology theory of Ozsv´ath and Szab´o [25]and Rasmussen [29] could prove this conjecture, and to make partial progress towardsimplementing this strategy. We will also try to provide evidence supporting the validityof our strategy.1.1. Background on the Berge Conjecture.
Before stating our results, we takesome time to review the conjecture. We begin by recalling Berge’s construction.
Construction B:
Let ( H , H , Σ) be the standard genus two Heegaard splitting of S .Here H , H are genus two handlebodies, joined along their common boundary ∂H = ∂H = Σ , a genus two surface. Let K ֒ → Σ be a knot embedded in the Heegaardsurface in such a way that ( ι k ) ∗ : π ( K ) ֒ → π ( H k ) ∼ = F represents a generator of thefundamental group of each handlebody, where ι k : Σ ֒ → H k , k = 1 , are the inclusion maps, and F is the free group on two generators. Following Berge, we call the knots in the above construction double primitive . Per-forming Dehn surgery on K with framing given by Σ can be thought of as attachinga pair of three-dimensional two-handles, A , A , to H and H , respectively, along K .The double primitive condition ensures that the resulting manifolds V i = H i ∪ A i havefundamental group Z . The loop theorem then implies each V i is a solid torus, and hencethe manifold obtained by the surgery is a lens space. We have the following conjecture,which is frequently referred to as the Berge conjecture. Berge Conjecture : If ( S , K ) is a knot on which Dehn surgery yields a lens space,then ( S , K ) is double primitive. Before proceeding, we make a few observations regarding the surgery slopes in theabove conjecture. Note first that while the conjecture makes no reference to the slopeof the surgery, the slope given by Construction B is clearly integral; it is specified bythe framing given by Σ. However, it follows from the Cyclic Surgery Theorem [7] thatunless ( S , K ) is a torus knot, any Dehn surgery on K yielding a lens space must beintegral. Since surgeries on torus knots yielding lens spaces are well-understood [17] wewill henceforth focus attention on integral surgeries unless otherwise specified.Evocative as the Berge Conjecture may be, for the purpose of explicitly enumeratingknots with lens space surgeries it is useful to view the problem from the perspective ofthe lens space. To do this, we first observe that a knot ( S , K ) on which Dehn surgeryyields the lens space L ( p, q ), naturally induces a knot ( L ( p, q ) , K ′ ). This knot is thecore of the solid torus glued to S − K in the surgery. Note that ( L ( p, q ) , K ′ ) is notnull-homologous, as it generates the first homology H ( L ( p, q ); Z ) ∼ = Z /p Z . Now it iseasy to see that ( L ( p, q ) , K ′ ) admits a surgery yielding the three-sphere: simply remove K ′ and undo the original surgery. Conversely, if surgery on ( L ( p, q ) , K ′ ) yields S , thereis an induced knot ( S , K ) on which surgery yields L ( p, q ). Thus the two perspectivesare equivalent.When studying knots in lens spaces admitting S surgeries, a natural class of knotsarises: Definition 1.1. ( One-bridge ) A knot ( L ( p, q ) , K ′ ) is called one-bridge with respectto the standard genus one Heegaard splitting of L ( p, q ) if K ′ is isotopic to a knot whichintersects each solid torus of the Heegaard splitting in a single unknotted arc. We will hereafter drop the Heegaard splitting from the terminology and simply saythat ( L ( p, q ) , K ′ ) is one-bridge. Such knots become relevant in light of Lemma 1 of[4], which shows that if ( S , K ) is double primitive, then the induced knot ( L ( p, q ) , K ′ )is one-bridge. A priori working with the class of one-bridge knots does not simplifymatters much. Indeed, there are clearly infinitely many one-bridge knots in L ( p, q );in particular, it contains torus knots - those knots which can be isotoped to lie in the N FLOER HOMOLOGY AND THE BERGE CONJECTURE 3
Heegaard torus - as a proper subset (see [6] for a classification scheme). However,amongst the one-bridge knots in L ( p, q ) is a particularly simple finite subfamily, whichwe call simple (or grid-number one ) knots. To describe these knots, let ( V α , V β , T ) bethe standard genus one Heegaard splitting of L ( p, q ), and let D α and D β be the meridiandiscs of the two solid tori, V α , V β . Assume that ∂D α , ∂D β have minimal intersectionnumber i.e. ∂D α ∩ ∂D β = { p pts } , see Figure 1. We then have Definition 1.2. ( Simple knot ) A one-bridge knot ( L ( p, q ) , K ′ ) is simple if either itbounds a disk, or is the union of two properly embedded arcs, t α , t β , in D α and D β ,respectively. See Figure 1. Note that there are p simple knots in L ( p, q ) - there is a unique simple knot in eachhomology class. For the reader familiar with Ozsv´ath-Szab´o Floer homology, simpleknots are precisely those knots which can be realized by placing two basepoints, z and w , on a minimally intersecting Heegaard diagram for L ( p, q ). For S there is a uniquesimple knot - the unknot - and it is the connection between knots in lens spaces and griddiagrams that motivates the alternate terminology grid-number one; simple knots arethose knots in lens spaces possessing a grid-diagram of grid-number one (see [16] and[3]). Simple knots and one-bridge knots are important for studying lens space surgeriesdue to the following theorem of Berge. Theorem 1.3. ( Theorem of [4]) Suppose ( L ( p, q ) , K ′ ) is a one-bridge knot whichadmits a three-sphere surgery. Then ( L ( p, q ) , K ′ ) is simple. We present a Floer homology proof of the above theorem in Section 3 (though in casethe knot induced in the three-sphere by the surgery satisfies p = 2 g ( K ) −
1, with g ( K )the Seifert genus, our theorem takes a slightly different form.) It is straightforward tosee that, upon performing the surgery on a simple knot ( L ( p, q ) , K ′ ), the induced knot( S , K ) is double primitive. We are thus led to the useful (equivalent) reformulation ofConjecture 1, which is stated as a question in [4]. Berge Conjecture : Suppose ( L ( p, q ) , K ′ ) is a knot which admits a three-spheresurgery. Then ( L ( p, q ) , K ′ ) is one-bridge. Coupled with Theorem 1.3, an affirmative answer to Conjecture 2 would allow oneto explicitly enumerate all knots in S on which surgery could yield a fixed lens space.To see this, assume surgery on ( S , K ) yields L ( p, q ). The above discussion showsthat the induced knot ( L ( p, q ) , K ′ ) has an S surgery and hence, by Conjecture 2 andTheorem 1.3, is simple. Now for each simple knot, K i , there is at most one integral slopesurgery producing an integer homology sphere, M ( K i ). Furthermore, the naturallypresented Heegaard splitting of M ( K i ) is genus two. One then uses the well-knownalgorithm to determine if a genus two Heegaard splitting is the three-sphere to determineif M ( K i ) ∼ = S . Each K i for which M ( K i ) ∼ = S has an induced (double primitive) knot MATTHEW HEDDEN
Figure 1.
Depiction of a simple knot K ′ in L (7 , L (7 ,
3) with minimal intersectionnumber ∂D α ∩ ∂D β . On it we have depicted a simple knot, K ′ , com-posed of two arcs t α and t β . Each arc can be isotoped along D α (resp. D β ) to a proper subinterval of ∂D α (resp. ∂D β ). Alternatively, K ′ couldbe specified by the two basepoints, z and w (see Definition 3.4 for thiscorrespondence). (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) PSfrag replacements ∂D α ∂D β t α t β z w in S on which surgery yields L ( p, q ). In this way, a proof of Conjecture 2 allows a finiteenumeration of knots for which surgery yields L ( p, q ).1.2. Statement of results - The role of Floer homology.
Using knot Floer ho-mology, we can hope to prove Conjecture 2. To do so recall that to any knot (
Y, K ) ina rational homology sphere (i.e. H ∗ ( Y ; Q ) ∼ = H ∗ ( S ; Q )), Ozsv´ath and Szab´o associatea collection of bigraded groups (see [28]), \ HF K ( Y, K ) := M ∗∈ Q , ξ ∈ Spin c ( Y,K ) \ HF K ∗ ( Y, K, ξ ) . These groups are graded by the Maslov index, which we denote by ∗ , and by relativeSpin c structures, ξ on Y − ν ( K ), the set of which we denote by Spin c ( Y, K ). The readerunfamiliar with relative Spin c structures can think of this as a grading by elementsof H ( Y − K ; Z ), since there is an affine isomorphism Spin c ( Y, K ) ∼ = H ( Y, K ; Z ) ∼ = H ( Y − K ; Z ). This grading should also be viewed as the analog of the Alexandergrading on the knot Floer homology of knots in S . That is, relative Spin c structuresplay the role of the powers of T in the Alexander polynomial of a knot ( S , K ). N FLOER HOMOLOGY AND THE BERGE CONJECTURE 5
Now the knot Floer homology groups of (
Y, K ) arise as the associated graded groupsof filtrations of the Ozsv´ath-Szab´o chain complexes d CF ( Y, s ) (here s is a Spin c structureon Y ). Thus there is a spectral sequence which begins with \ HF K ( Y, K ) and convergesto d HF ( Y ), where d HF ( Y ) is the direct sum: d HF ( Y ) := M s ∈ Spin c ( Y ) d HF ( Y, s ) . It follows immediately that we have the inequality of ranks:rk d HF ( Y ) ≤ rk \ HF K ( Y, K )We say that a knot has simple
Floer homology if equality holds. In the case of lensspaces, rk d HF ( L ( p, q ) , s ) = 1 for every s ∈ Spin c ( L ( p, q )). Thus ( L ( p, q ) , K ′ ) has simpleFloer homology if rk \ HF K ( L ( p, q ) , K ′ ) = p .Our first step towards Conjecture 2 is the following restriction on the knot Floerhomology of the knot in L ( p, q ) induced by the surgery. Theorem 1.4.
Let ( S , K ) be a knot of Seifert genus g and suppose that there exists aninteger p > such that p surgery on K yields the lens space, L ( p, q ) . Let ( L ( p, q ) , K ′ ) be the knot induced by the surgery. Then (1) p ≥ g − , (2) If p ≥ g , then ( L ( p, q ) , K ′ ) has simple Floer homology, (3) If p = 2 g − , then rk \ HF K ( L ( p, q ) , K ′ ) = rk d HF ( L ( p, q )) + 2 . Remark 1.5.
By reflecting K if necessary, the assumption that p be positive is non-restrictive. This theorem was recently proved by Rasmussen [31] using a different strat-egy. The fact that p ≥ g − p = 2 g − Y , with rk d HF ( Y, s ) = 1 for each s ∈ Spin c ( Y ).) The Berge conjectures wouldthen follow from Conjecture 1.6. (Conjecture . of [3] ) A knot ( L ( p, q ) , K ′ ) with simple Floer homologyis simple (in the sense of Definition 1.2. Conjecture 1.7. (Conjecture . of [3] ) There are exactly two knots T R , T L ⊂ L ( p, q ) which satisfy rk \ HF K ( L ( p, q ) , T ) = rk d HF ( L ( p, q )) + 2 . MATTHEW HEDDEN
In Section 3, two knots satisfying rk ( \ HF K ( L ( p, q ) , K )) = p + 2 are specified for eachlens space and we show that surgery on them cannot produce S . Thus a proof of theabove conjectures, together with Theorem 1.4, would indeed prove the Berge conjecture.Note that the hypothesis for these conjectures only involves the total rank of the knotFloer homology groups of ( L ( p, q ) , K ′ ). The groups have a rich structure inherited fromtheir bigrading. It is possible that it would be necessary to exploit this structure. Thuswe are also led to: Conjecture 1.8.
Let ( L ( p, q ) , G ) be any simple knot. Suppose that for some knot ( L ( p, q ) , K ′ ) , we have \ HF K ∗ ( L ( p, q ) , K ′ , s i ) ∼ = \ HF K ∗ ( L ( p, q ) , G, s ) , for all ∗ and s . Then ( L ( p, q ) , K ′ ) is isotopic to ( L ( p, q ) , G ) . That is, simple knots arecharacterized by their Floer homology. In Section 4 we provide some justification for the conjectures. In particular, by usingwork of Baker [1], we prove Conjectures 1.6 and 1.7 for knots in L ( p, q ) satisfying a genusconstraint. To describe this constraint, let us consider only those knots ( L ( p, q ) , K ′ )whose homology class [ K ′ ] ∈ H ( L ( p, q ); Z ) generates. For such a knot it makes senseto define the genus of K ′ , denoted g ( K ′ ), to be the minimum genus of any properlyembedded surface-with-boundary i : ( F, ∂F ) ֒ → ( L ( p, q ) − νK ′ , ∂νK ′ )whose homology class is Poincar´e dual to the generator of H ( L ( p, q ) − K ′ ; Z ) ∼ = Z . Wethen have Theorem 1.9.
Let ( L ( p, q ) , K ′ ) be any knot whose homology class generates H ( L ( p, q ); Z ) and which satisfies g ( K ′ ) ≤ p + 14 . Then Conjectures 1.6 and 1.7 hold for ( L ( p, q ) , K ′ ) . In particular, if ( L ( p, q ) , K ′ ) hassimple Floer homology then ( L ( p, q ) , K ′ ) is simple. Moreover, by using Baker’s result with a result of Ni [18], it is also possible to proveConjecture 1.8 for an infinite family of simple knots. This result seems quite interestingin its own right, as it provides the first infinite family of knots with non-trivial Thurstonnorm which are characterized by their knot Floer homology (the previous known exam-ples being the figure-eight and trefoil knots). We briefly discuss this result in Section 4and postpone the detailed proof for later [13].
Outline:
The next section provides the proof of Theorem 1.4 which relies heavily onprevious work of the author and Ozsv´ath and Szab´o. Aided by this theorem, Section 3uses a simple Floer homology argument to prove Berge’s Theorem 2, mentioned above.While our argument uses the machinery of Floer homology, it avoids the use of the
N FLOER HOMOLOGY AND THE BERGE CONJECTURE 7 algorithm to detect if a genus two Heegaard splitting is the three-sphere and the CyclicSurgery Theorem [7]. In the final section we discuss evidence for the conjectures andprove Theorem 1.9.
Acknowledgments:
This work has benefited much from conversations with Ken Bakerand Eli Grigsby, and the general strategy presented here is part of our joint ongoingwork [3]. I also thank Cameron Gordon for generous sharing of his knowledge of Dehnsurgery, and Jake Rasmussen for sharing his independent work on Floer homology andthe Berge conjecture.
MATTHEW HEDDEN Proof of Main Theorem
Outline.
This section is devoted to a proof of Theorem 1.4. Before beginning,we briefly sketch the idea. Denote the p -twisted (positive-clasped) Whitehead doubleof a knot K ֒ → S by D + ( K, p ) (this is a specific type of satellite knot, see Figure 2).A formula for the knot Floer homology of D + ( K, p ) was exhibited in Theorem 1 . K . A key step in the proof of the formula was an iden-tification of a particular Floer homology group associated to D + ( K, p ) with the directsum of all the Floer homology groups of the induced knot ( S p ( K ) , K ′ ). Here, S p ( K )denotes the manifold obtained by p -surgery on K , and K ′ denotes the knot induced bythe surgery i.e. the core of the solid torus glued to S − K in the surgery. Knowingthis identification, we can apply the formula for the Floer homology of the Whiteheaddouble to calculate the total rank of the Floer homology groups of ( S p ( K ) , K ′ ). In thespecial case that S p ( K ) is the lens space L ( p, q ), Ozsv´ath and Szab´o have an explicitformula (Theorem 1 . K in terms of the Alexander polynomial of K . Combining thesetwo theorems, Theorem 1.4 will follow readily. As with Ozsv´ath and Szab´o’s Theorem,our theorem will handle the more general situation when S p ( K ) is an L -space, ratherthan a lens space (recall that an L -space is a rational homology sphere Y for whichrk d HF ( Y, s ) = 1 for every s ∈ Spin c ( Y )): Theorem 1.4
Let ( S , K ) be a knot and suppose that there exists an integer p > such that S p ( K ) is an L -space. Let ( S p ( K ) , K ′ ) be the induced knot. Then (1) p ≥ g − , (2) If p ≥ g , then rk \ HF K ( S p ( K ) , K ′ ) = rk d HF ( S p ( K )) = p , (3) If p = 2 g − , then rk \ HF K ( S p ( K ) , K ′ ) = rk d HF ( S p ( K )) + 2 . Proof of Theorem 1.4.
We begin the details of our proof. We first note that,by deferring to [14], we can immediately dispatch with part (1) of the theorem. Indeedthe fact that p ≥ g − . . F will denote the field Z / Z ,and F ( l ) will indicate this same field endowed with Maslov grading l . N FLOER HOMOLOGY AND THE BERGE CONJECTURE 9
Figure 2.
The positive t -twisted Whitehead double, D + ( K, p ), of theleft-handed trefoil. Start with a twist knot, P , with t full twists embeddedin a solid torus, V . The “ + ” indicates the parity of the clasp of P . f identifies V with the neighborhood of K , νK , in such a way that thelongitude for V is identified with the Seifert framing of K . The imageof P under this identification is D + ( K, p ). The 3 extra full twists in theprojection of D + ( K, p ) shown arise from the writhe of the trefoil, − t+3 V P νKD + ( K, p )1 f+=
Theorem 2.1.
Let ( S , K ) be a knot and let ( S p ( K ) , K ′ ) be the knot induced in S p ( K ) by the core of the surgery torus. Then \ HF K ( D + ( K, p ) , ∼ = M { s ∈ Spin c ( S p ( K ) ,K ′ ) } \ HF K ( S p ( K ) , K ′ , s ) , Remark 2.2.
The statement above differs from that found in [12] in two ways. First, thestatement presented in [12] expresses the knot ( S p ( K ) , K ′ ) as ( S p ( K ) , µ K ) , where µ K isthe meridian of the knot ( S , K ) viewed as knot in S p ( K ) . However, it is straightforwardto see that in S p ( K ) , µ K is isotopic to K ′ - one simply uses the meridian disc of thesurgery torus to perform the isotopy. Second, the right hand side of the congruenceis a sum over relative Spin c structures, instead of a double sum over absolute Spin c structures on S p ( K ) and filtration levels induced by µ K . However, as mentioned in the introduction the filtration of d CF ( Y, s ) induced by µ K is by relative Spin c structureswhich s extends, and thus the single sum above is equivalent. Next, we have Theorem 1 . S isthe Ozsv´ath-Szab´o “hat” chain complex, d CF ( S ), and that the homology of this chaincomplex is given by d HF ( S ) ∼ = F (0) . Next, recall that to a knot ( S , K ) Ozsv´ath andSzab´o [25] associate a filtered version of the chain complex, d CF ( S ) (see also [ ? ]). Thatis, we have an increasing sequence of subcomplexes:0 = F ( K, − i ) ֒ → F ( K, − i + 1) ֒ → . . . ֒ → F ( K, n ) = d CF ( S ) . The filtered chain homotopy type of this filtration is an invariant of the pair, ( S , K ).We denote the quotient complexes F ( K,j ) F ( K,j − := \ CF K ( K, j ). The homology of thesequotients, denoted \ HF K ( K, j ), are the knot Floer homology groups of K . As in [23],we define: τ ( K ) = min { j ∈ Z | i ∗ : H ∗ ( F ( K, j )) −→ H ∗ ( d CF ( S )) is non-trivial } . This number is the Ozsv´ath-Szab´o concordance invariant which, as its name suggests,has been useful in the study of smooth knot concordance, see [23, 15]. In terms of theseinvariants we have
Theorem 2.3. (Theorem . of [12] ) Let ( S , K ) be a knot with Seifert genus g ( K ) = g .Then for p ≥ τ ( K ) we have: \ HF K ( D + ( K, p ) , ∼ = F p − g − g M i = − g [ H ( F ( K, i ))] , Whereas for p < τ ( K ) the following holds: \ HF K ( D + ( K, p ) , ∼ = F − p +4 τ ( K ) − g − g M i = − g [ H ( F ( K, i ))] . Stated above is only the part of Theorem 1 . \ HF K ( D + ( K, p ) , F − p +4 τ ( K ) − g − this exponent could verywell be negative. By F − n , for instance, we mean the quotient of the remaining groupby a subgroup of dimension n .Let us now recall Theorem 1 . N FLOER HOMOLOGY AND THE BERGE CONJECTURE 11
Theorem 2.4.
Let ( S , K ) be a knot of Seifert genus g ( K ) = g and suppose that thereexists an integer p > such that S p ( K ) is an L -space. Then there is an increasingsequence of integers − g = n − k < ... < n k = g with the property that n i = − n − i , and the following significance. If for − g ≤ i ≤ g welet δ i = if i = gδ i +1 − n i +1 − n i ) + 1 if g − i is odd δ i +1 − if g − i > is even,then \ HF K ( K, j ) = 0 unless j = n i for some i , in which case \ HF K ( K, j ) ∼ = Z and itis supported entirely in homological dimension δ i . We will use Theorem 2.4 together with Theorems 2.3 and 2.1 to deduce Theorem 1.4.First observe that Theorem 2.4 determines the invariant τ ( K ) for knots admitting L -space surgeries: Corollary 2.5. (Corollary . of [24] ) Let ( S , K ) be a knot of Seifert genus g ( K ) and suppose that there exists an integer p > such that S p ( K ) is an L -space. Then τ ( K ) = g ( K ) . Proof.
This follows immediately from the description of the knot Floer homologygroups of K given by Theorem 2.4 and the definition of τ ( K ). In particular, the onlyknot Floer homology group supported in homological grading 0 is in filtration grading g ( K ).From this corollary, we can insert g ( K ) in place of τ ( K ) in Theorem 2.3 and combinethe result with Theorem 2.1 to yield Proposition 2.6.
Let ( S , K ) be a knot with Seifert genus g ( K ) = g , and suppose thatthere exists an integer p > such that S p ( K ) is an L -space. Then for p ≥ g we have: M { s ∈ Spin c ( S p ( K ) ,K ′ ) } \ HF K ( S p ( K ) , K ′ , s ) ∼ = F p − g − g M i = − g − [ H ( F ( K, i )) ⊕ H ( F ( K, − i − , Whereas for p < g the following holds: M { s ∈ Spin c ( S p ( K ) ,K ′ ) } \ HF K ( S p ( K ) , K ′ , s ) ∼ = F − p +2 g − g M i = − g − [ H ( F ( K, i )) ⊕ H ( F ( K, − i − , Note that we have chosen to rewrite the direct sum on the far right of the aboveformulas in a slightly different form. To see the equivalence, note that the adjunctioninequality for knot Floer homology (Theorem 5 . H ∗ ( F ( K, j )) ∼ = 0whenever j < − g ( K ). Thus we have: g M i = − g [ H ( F ( K, i ))] ∼ = g M i = − g − [ H ( F ( K, i ))] . Now it is easy to rewrite the right hand side as it appears in the proposition: g M i = − g − [ H ( F ( K, i ))] = g M i = − g − [ H ( F ( K, i )) ⊕ H ( F ( K, − i − . The motivation for the manipulation is the following lemma, which shows that the rankof [ H ( F ( K, i )) ⊕ H ( F ( K, − i − L -space surgeries. Proposition 2.7.
Let ( S , K ) be a knot, and suppose that there exists an integer p > such that S p ( K ) is an L -space. Then rk H ( F ( K, m )) + rk H ( F ( K, − m − , for all m. Proof.
The proof relies on a Theorem of Ozsv´ath and Szab´o which relates the Floerhomology of S p ( K ) to the Floer homology of the filtration induced on the “infinity”chain complex of the three-sphere, CF ∞ ( S ), by the knot. More precisely, there is arefined version of the knot Floer homology filtration described above which associatesto a knot ( S , K ) a Z ⊕ Z -filtered chain complex, CF K ∞ ( S , K ). Generators of thischain complex correspond to triples, [ x , i, j ], where x ∈ T α ∩ T β is an intersection pointof two “lagrangian” tori in the symmetric product of a Heegaard surface for S , and i, j ∈ Z satisfy a homotopy theoretic constraint: h c ( s ( x )) , [Σ] i + 2( i − j ) = 0The above constraint is described by the following: Ozsv´ath and Szab´o assign a relativeSpin c structure, s ∈ Spin c ( S − K ), to x ∈ T α ∩ T β . In the present case relative Spin c structures can be canonically identified with Spin c structures on S ( K ) and, as such,have a well-defined first Chern class, c ( s ( x )) which can be evaluated on the generator[Σ] of H ( S ( K ); Z ) ∼ = Z .By construction, the generators of CF K ∞ ( S , K ) admit a map F : CF K ∞ ( S , K ) → Z ⊕ Z , determined by F ([ x, i, j ]) = ( i, j ). If we define a partial ordering on Z ⊕ Z by ( i, j ) ≤ ( i ′ , j ′ ) if i ≤ i ′ and j ≤ j ′ , then F is a filtration i.e. F ( ∂ ∞ [ x, i, j ]) ≤ F ([ x, i, j ]), where ∂ ∞ is the differential on CF K ∞ ( S , K ) (see [20, 25] for a definition of this differential).The rich algebraic structure inherent in a doubly filtered chain complex allows one to N FLOER HOMOLOGY AND THE BERGE CONJECTURE 13 examine the homology of many subobjects defined by generators whose filtration indicessatisfy specific numerical constraints. For instance, one can define the chain complex: C { i = 0 } ⊂ CF K ∞ ( S , K ) , consisting of generators of the form [ x , , j ] for some j ∈ Z . This set naturally inheritsa differential from CF K ∞ ( S , K ), since it is a subcomplex of the quotient complex CF K ∞ C { i< } . We have the chain homotopy equivalence of chain complexes C { i = 0 } ≃ d CF ( S ) . Thus we recover the Ozsv´ath-Szab´o “hat” complex of S . Furthermore, we have: F ( K, m ) ≃ C { i = 0 , j ≤ m } , and by the filtered chain homotopy equivalence between the filtration associated to aknot K and its reverse, − K (Proposition 3 . F ( K, − m − ≃ C { i< , j = m } . (note that this chain homotopy equivalence does not preserve the Maslov grading, butas the proposition does not reference the Maslov grading we do not belabor this point.)Perhaps the most important aspect of CF K ∞ ( S , K ) chain complex is that it deter-mines the Floer homology groups of manifolds obtained by Dehn surgery on K . Indeed,Theorem 4 . c structures on S p ( K )by elements [ m ] ∈ Z /p Z ↔ Spin c ( S p ( K )) we have a chain homotopy equivalence, H ∗ ( C { max( i, j − m ) = 0 } ) ∼ = d HF ( S p ( K ) , s [ m ] ) , for all p ≥ g ( K ) −
1. Now we have the following short exact sequence of chaincomplexes:0 −−−→ C { i< , j = m } i −−−→ C { max( i, j − m ) = 0 } p −−−→ C { i = 0 , j ≤ m } −−−→ . . . −−−→ H ( F ( K, − m − i ∗ −−−→ d HF ( S p ( K ) , s [ m ] ) p ∗ −−−→ H ( F ( K, m )) −−−→ . . . Under the assumption that S p ( K ) is an L-space, the middle term has rank one foreach m . The proposition will now follow from exactness of the above sequence and thefollowing: Claim 2.8.
Let
K ֒ → S be a knot and suppose that p > surgery on K yields anL-space. Then rk H ( F ( K, m )) ≤ , for all m . We prove the claim with the help of Theorem 2.4 above. To do this, recall that theknot Floer homology groups can be viewed as a filtered chain complex in their ownright, endowed with a differential which strictly lowers the filtration grading. This ismade precise by Lemma 4 . Lemma 2.9. (Lemma . of [29] ) Let C be a filtered complex with filtration C ⊂ C ⊂ . . . ⊂ C m , and let C i = C i /C i − be the filtered quotients, so that the homology groups H ∗ ( C i ) arethe E terms of the spectral sequence associated to the filtration. Then, up to filteredchain homotopy equivalence, there is a unique filtered complex C ′ with the followingproperties: (1) C ′ is filtered chain homotopy equivalent to C . (2) ( C ′ ) i ∼ = H ∗ ( C i )(3) The spectral sequence of the filtration on C ′ has trivial first differential. Itshigher terms are the same as the higher terms of the spectral sequence of thefiltration on C . In the present situation the lemma allows us to replace the filtered chain complex( \ CF K ( K ) , ∂ ) by ( \ HF K ( K ) , ∂ ′ ). Here, \ HF K ( K ) is meant to indicate the direct sumof the knot Floer homology groups of K , ⊕ \ HF K ( K, i ). In light of this, we have theisomorphisms of Maslov graded groups: H ∗ ( F ( K, m )) ∼ = H ∗ ( ⊕ i ≤ m \ HF K ( K, i ) , ∂ ′ ) ,H ∗ ( \ HF K ( K ) , ∂ ′ ) ∼ = H ∗ ( \ CF K ( K ) , ∂ ) ∼ = d HF ( S ) ∼ = F (0) . With this algebraic aside behind us, let us return to the proof of the claim. Assumethen, that rk H ∗ ( F ( K, m )) > m . Since S p ( K ) is an L -space, Theorem 2.4indicates that rk \ HF K ( K, j ) ≤ j. Furthermore, the Maslov gradings of the non-trivial groups are a strictly increasingfunction of the filtration grading, with maximum Maslov grading 0. This, togetherwith our assumption that rk H ∗ ( F ( K, m )) > G ⊂ H ∗ ( F ( K, m )) satisfying G ∼ = F ( i ) ⊕ F ( k ) , i < k ≤ . Using again the fact that the Maslov gradings of the non-trivial Floer homology groupsare a strictly increasing function of the filtration grading we have ⊕ j>m \ HF K i +1 ( K, j ) ∼ = 0 . This in turn implies that the summand F ( i ) ⊂ G survives under the induced differentialon \ HF K ( K ) - there are simply no chains in Maslov grading i +1 to map to the generator N FLOER HOMOLOGY AND THE BERGE CONJECTURE 15 of F ( i ) . Survival of the F ( i ) summand, however, contradicts the fact that d HF ( S ) ∼ = F (0) .Thus rk H ∗ ( F ( K, m ) ≤ m as claimed.We now complete the proof of Theorem 1.4. Propositions 2.6 and 2.7 show that for p ≥ g ( K ) we have:Σ s rk \ HF K ( S p ( K ) , K ′ , s ) = p − g − g Σ − g − p = rk d HF ( S p ( K )) , whereas for p = 2 g ( K ) − s rk \ HF K ( S p ( K ) , K ′ , s ) = 2 g − − p + g Σ − g −
1= 2 g − − p + 2 g + 2 = 4 g − p = 2 g + 1 = rk d HF ( S p ( K )) + 2 . A Floer homology proof of Berge’s theorem
Recall from the introduction that a that a knot ( L ( p, q ) , K ′ ) is called zero-bridge withrespect to the standard genus one Heegaard splitting of L ( p, q ) if K ′ is isotopic to a knotlying in the torus of the splitting. A knot ( L ( p, q ) , K ′ ) is called one-bridge with respectto the standard genus one Heegaard splitting of L ( p, q ) if K ′ is isotopic to a knot whichintersects each solid torus of the Heegaard splitting in a single unknotted arc.In this section we will use Theorem 1.4 to prove the following: Theorem 3.1.
Suppose ( L ( p, q ) , K ′ ) is a one-bridge and that ( L ( p, q ) , K ′ ) admits athree-sphere surgery. Let K be the knot in S induced by the surgery, and let g denoteits Seifert genus. Then either • p > g − , in which case ( L ( p, q ) , K ′ ) is simple • p = 2 g ( K ) − and ( L ( p, q ) , K ′ ) is one of the two knots, T R , T L specified byFigure 3 (or their reversals). We remark that our theorem is more general: it holds for any one-bridge knot withan integral homology sphere L-space surgery. We further note that from our theoremwe immediately recover Berge’s theorem (Theorem 2 of [4]) in the case p > g − p = 2 g − Figure 3.
Doubly-pointed Heegaard diagrams specifying the two knotsreferenced in Theorem 3.1 and Conjecture 1.7. Shown are T R and T L inthe lens space L (7 , S , T R and T L are the right- and left-handedtrefoils, respectively. In general, a Heegaard diagram for T R and T L isobtained from the minimal diagram of L ( p, q ) be a simple isotopy whichcreates 2 extra intersection points. Z WZ W (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
PSfrag replacements ∂D α ∂D β t α t β T L T R N FLOER HOMOLOGY AND THE BERGE CONJECTURE 17 theorem tells us that a one-bridge knot in L ( p, q ) with a three-sphere surgery is simple,independent of p . Thus in the case p = 2 g − Corollary 3.2.
Let ( L ( p, q ) , K ′ ) be a one-bridge knot which admits a three-spheresurgery. Let K be the knot in S induced by the surgery, and g denote its Seifertgenus. Then p > g − . Proof. If p = 2 g −
1, then ( L ( p, q ) , K ′ ) is not simple by the above theorem, contra-dicting Berge’s result.To prove our theorem, we first note that since ( L ( p, q ) , K ′ ) admits a three-spheresurgery, the induced knot in S has a lens space surgery. Thus Theorem 1.4 appliesand we see that either ( L ( p, q ) , K ′ ) has simple Floer homology or p = 2 g − \ HF K ( L ( p, q ) , K ′ ) = rk d HF ( L ( p, q )) + 2 = p + 2. In light of this, the followingproposition implies the theorem stated above. Proposition 3.3.
Let ( L ( p, q ) , K ′ ) be a one-bridge knot. • If ( L ( p, q ) , K ′ ) has simple Floer homology then ( L ( p, q ) , K ′ ) is simple. • If rk \ HF K ( L ( p, q ) , K ′ ) = p + 2 then ( L ( p, q ) , K ′ ) is one of the knots specifiedby Figure 3 (or their reversals). Knot Floer homology background for one-bridge knots.
Before proving theproposition, we collect some basic facts and definitions surrounding the calculation ofknot Floer homology for one-bridge knots in lens spaces. First recall the definition of acompatible doubly-pointed Heegaard diagram for a knot, (
Y, K ): Definition 3.4. A compatible doubly-pointed Heegaard diagram for a knot ( Y, K ) (orsimply a Heegaard diagram for ( Y, K ) ) is a collection of data (Σ , { α , . . . , α g } , { β , . . . , β g } , w, z ) , where • Σ is an oriented surface of genus g, the Heegaard surface , • { α , . . . , α g } are pairwise disjoint, linearly independent embedded circles (the α attaching circles ) which specify a handlebody, U α , bounded by Σ , • { β , . . . , β g } are pairwise disjoint, linearly independent embedded circles whichspecify a handlebody, U β , bounded by Σ such that U α ∪ Σ U β is diffeomorphic to Y , • K is isotopic to the union of two arcs joined along their common endpoints w and z . These arcs, t α and t β , are properly embedded and unknotted in the α and β -handlebodies, respectively. See Figure 3 for an example. It was first pointed out in Proposition 2 . S are precisely those knots admitting a genus one Heegaard diagram (note that [11]refers to one-bridge knots as (1 ,
1) knots). Two points are worth mentioning here. Thefirst is that Proposition 2 . S . However, it follows from Condition (iii) in thestatement of Proposition 2 . K (thefirst β attaching curve) intersects exactly one attaching curve for the α handlebody inexactly one point - that after performing handleslides and isotopies of the attachingcurves the Heegaard diagram can be destabilized to a genus one diagram. Indeed, thisobservation was exploited in [11] throughout the discussion. The second observationis that Proposition 2 . L ( p, q ) that are at most one-bridge admit genus one Heegaarddiagrams.In Section 6 of [25], Ozsv´ath and Szab´o develop a general technique for calculating theFloer homology of any knot admitting a genus one Heegaard diagram (again, this wasonly explicitly stated for Y = S but holds for any lens space). Very briefly, recall thatknot Floer homology was first defined as the “Lagrangian” intersection Floer homologyfor two totally real submanifolds in the g -fold symmetric product of the Heegaard surface(with an appropriate almost complex structure). These totally real submanifolds aredefined by the attaching curves of the Heegaard diagram. However, in the case at hand- when we are dealing with a genus one Heegaard surface - the construction can bedescribed much more concisely.Given a genus one Heegaard diagram for a knot ( L ( p, q ) , K ′ ),( T , α, β, w, z ) , we construct a chain complex \ CF K ( L ( p, q ) , K ′ ) as follows. The generators are inter-section points of the attaching curves: \ CF K ( L ( p, q ) , K ′ ) = ⊕ x ∈ α ∩ β F · < x > . Here, for simplicity, we take coefficients in the field with two elements, F = Z / Z . Todefine the boundary operator, let us associate an incidence number to x, y ∈ α ∩ β asfollows. To begin let π ( x, y ) denote the set of homotopy classes of maps of 5-tuples: u : ( D , e α , e β , − i, i ) → ( T , α, β, x, y ) , where D ⊂ C is the unit disc, e α (resp. e β ) is the part of its boundary with positive(resp. negative) real part, i = √−
1, and the right hand side is the Heegaard diagram.We set n ( x, y ) = − preserving u ∈ π ( x, y )with no obtuse corners and z, w / ∈ Im ( u )0 otherwise N FLOER HOMOLOGY AND THE BERGE CONJECTURE 19
Figure 4.
Depiction of disks in π ( x, y ) counted in the incidence number n ( x, y ). Shown are two maps of disks, u and u ′ . Both satisfy the boundaryconditions necessary to be in π ( x, y ) (resp. π ( x ′ , y ′ )), and both areorientation preserving. Because the image of u ′ has an obtuse corner, itis not counted in the incidence number n ( x ′ , y ′ ). (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) PSfrag replacements C e α e β i − i x yu x ′ y ′ u ′ See Figure 4 for an explanation of these conditions. In terms of these incidence numbers,the boundary operator can be described by: ∂x = Σ y ∈ α ∩ β n ( x, y ) y The knot Floer homology groups are the homology groups of this chain complex: \ HF K ( L ( p, q ) , K ′ ) := H ∗ ( \ CF K ( L ( p, q ) , K ′ ) , ∂ ) . For our present purposes, we make no reference to the gradings of this group, butremark that there is a bigrading on the generators of \ CF K ( L ( p, q ) , K ′ ) coming fromthe Maslov index and relative Spin c structures on L ( p, q ) − K ′ . We will only have needfor the total rank of the knot Floer homology groups, summing over both gradings.3.2. Proof of Proposition 3.3.
With the above preliminaries behind us, the Propo-sition will follow quickly.
Proof.
We handle first the case when ( L ( p, q ) , K ′ ) is a one-bridge knot with simpleFloer homology. First observe that simple knots are exactly those knots which canbe described by doubly-pointed genus one Heegaard diagrams with minimal intersec-tion number between the α and β curves i.e. ( L ( p, q ) , K ′ ) is simple if and only if ithas a genus-one doubly pointed Heegaard diagram with exactly p intersection points.Now suppose that ( L ( p, q ) , K ′ ) is one-bridge but not simple. Let ( T , α, β, z, w ) be acompatible doubly-pointed Heegaard diagram for ( L ( p, q ) , K ′ ) with the fewest num-ber of intersection points x ∈ α ∩ β . Since ( L ( p, q ) , K ′ ) is not simple we must have geom | α ∩ β | > p . Thus, if ( L ( p, q ) , K ′ ) has simple Floer homology we haverk \ CF K ( L ( p, q ) , K ′ ) > rk \ HF K ( L ( p, q ) , K ′ ) = p. In order for the above inequality to hold there must exist a pair of generators, x, y , forwhich n ( x, y ) = 1. This implies the existence of a map from a disc u , as above, whoseimage misses both basepoints z, w defining the Heegaard diagram. We can use thisdisc, as in Figure 5, to remove both intersection points (strictly speaking, the isotopymay remove multiple intersection points depending on whether there are other discspresent). In this way we arrive at a Heegaard diagram with strictly fewer intersectionpoints than the one we started with, contradicting our assumption that ( T , α, β, z, w )was minimal with respect to geometric intersection number.The case when rk \ HF K ( L ( p, q ) , K ′ ) = p + 2 is only slightly more involved. As above,let ( T , α, β, z, w ) be a compatible doubly-pointed Heegaard diagram for ( L ( p, q ) , K ′ )with the fewest number of intersection points x ∈ α ∩ β . By the isotopy argumentdescribed above, we can assume this number to be p + 2. That is(1) rk \ CF K ( L ( p, q ) , K ′ ) = rk \ HF K ( L ( p, q ) , K ′ ) = p + 2 . Figure 5.
If there exists a non-trivial differential for the chain complex d CF ( L ( p, q, K ′ ), then the Heegaard diagram for K can be simplified by anisotopy. (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) N FLOER HOMOLOGY AND THE BERGE CONJECTURE 21
To show that ( L ( p, q ) , K ′ ) is the knot of Figure 3, we note that there is a refinedincidence number: n z ( x, y ) = − preserving u ∈ π ( x, y )with no obtuse corners and w / ∈ Im ( u )0 otherwiseand that the operator defined by ∂ z x = Σ y ∈ α ∩ β n z ( x, y ) y satisfies ∂ z ◦ ∂ z = 0. Further, the homology of the resulting chain complex satisfies(2) rk H ∗ ( \ CF K ( L ( p, q ) , K ′ ) , ∂ z ) = p. Similar remarks hold if we switch the roles of z and w . That is, there exists an analogousboundary operator ∂ w on \ CF K ( L ( p, q ) , K ′ ) whose homology is also of rank p . NowEquations (1),(2) imply the existence of intersection points x, y ∈ α ∩ β , and a disk u ∈ π ( x, y ) which satisfies: • u is orientation preserving • Im( u ) has no obtuse corners • alg z ∩ Im( u ) = geom z ∩ Im( u ) ≥ u with the submanifold z ֒ → T (the fact that the two numbers are equal followsfrom the fact that u is orientation preserving). We claim that in fact, z ∩ Im( u ) = 1.Indeed, if this were not the case, we could lift the Heegaard diagram and Im( u ) to theuniversal cover of T to show that there also exist intersection points x ′ , y ′ in α ∩ β and adisk u ∈ π ( x ′ , y ′ ) satisfying z ∩ Im( u ) = 1 (and also the other two conditions itemizedabove). This, in turn, implies that geom | α ∩ β | > p + 2, contradicting our assumption.Using ∂ w , a similar discussion shows the existence of intersection points r, s ∈ α ∩ β ,and a disk v ∈ π ( r, s ) satisfying w ∩ Im( v ) = 1The proposition now follows from the claim that, of the four intersection points x, y, r, s , two must be equal: for if two of the intersection points are equal, then theexistence of the disks u and v force the Heegaard diagram for ( L ( p, q ) , K ′ ) to takethe form of Figure 3. However, the claim follows from the observation that if none of x, y, r, s are equal, geom | α ∩ β | > p + 2. Intuition for Conjectures 1.6-1.7 and proof of Theorem 1.9
The purpose of this Section is to provide some justification for why the conjecturescited in the introduction would be true. Loosely speaking, Conjecture 1.6 and 1.7 saythat simple Floer homology implies simple knot. Conjecture 1.8 says that simple knotsare characterized by their Floer homology.We first point out that all three conjectures hold in the case where L ( p, q ) = S . Thisis made precise by the theorems of Ozsv´ath and Szab´o and Ghiggini: Theorem 4.1. (Theorem . of [26] ) Suppose K ⊂ S satisfies rk ( \ HF K ( S , K )) = 1 .Then K is the unknot (the only simple knot in S ). Theorem 4.2. (Corollary . of [9] ) Suppose K ⊂ S satisfies rk ( \ HF K ( S , K )) = 3 .Then K is the right- or left-handed trefoil. For knots in a general lens space, perhaps the most compelling quantitative evidenceat the moment is Theorem 1.9, which says that Conjectures 1.6 and 1.7 hold for knotswhose complements have somewhat simple topology.
Proof of Theorem 1.9.
Suppose that ( L ( p, q ) , K ′ ) is a knot whose homology classgenerates H ( L ( p, q ); Z ) and for which(3) g ( K ′ ) ≤ p + 14 . Theorem 1 . L ( p, q ) , K ′ ) is one-bridge. Now Proposition 3.3 appliesand shows that if ( L ( p, q ) , K ′ ) has simple Floer homology then ( L ( p, q ) , K ′ ) is simple.Strictly speaking, we should also address the case when rk \ HF K ( L ( p, q ) , K ′ ) = p + 2.However, in this case any knot satisfying the genus constraint would be one-bridge and,by Proposition 3.3, would be the knot of Figure 3. However, it can be shown (using, forexample, results of Ni [18]) that this knot does not possess surfaces in its complementwhich satisfy the genus constraint. (cid:3) For knots in L ( p, q ) which satisfy Equation (3), one can also prove Conjecture 1.8.To describe the method by which this is done, assume that we are given a simple knot( L ( p, q ) , K ′ ) satisfying (3). The calculation of the bigraded Floer homology groups ofany simple knot is straightforward. Now a theorem of Ni [18] shows that the breadthof the homology support in the Alexander grading determines the genus of any knot( L ( p, q ) , J ). Thus, if another knot ( L ( p, q ) , J ) had the same Floer homology as a simpleknot satisfying (3), J would necessarily satisfy (3) as well. By Baker’s theorem, thiswould imply that J is one-bridge and Proposition 3.3 would imply that J is simple. Thenone can check that there is a unique simple knot in L ( p, q ) with the Floer homology of( L ( p, q ) , K ′ ). We postpone the details of this argument for a later time, but suffice it tosay that it is straightforward to find infinite families of simple knots (in different lensspaces) of arbitrarily large genus which are characterized by their knot Floer homology. N FLOER HOMOLOGY AND THE BERGE CONJECTURE 23
In another direction, the recent connection between knot Floer homology and griddiagrams [16] also provides compelling evidence for Conjecture 3. In [16] it was shownthat the Floer homology of knots in S can be combinatorially computed from a certain(grid) diagram associated to a particular (grid) projection of K . The extension ofthis combinatorial formula to knots in lens spaces is discussed in [3]. There, a similarformula to that for knots in S is presented which computes the knot Floer homologyof an arbitrary knot ( L ( p, q ) , K ′ ). This formula, too, is in terms of a grid diagramfor ( L ( p, q ) , K ′ ). To date, there are no combinatorial proofs of Theorems 4.1 and 4.2.The existing proofs rely on connections between Ozsv´ath-Szab´o theory and symplecticgeometry. However, it seems reasonable to expect that if either or both of these theoremscould be understood combinatorially, then the proofs could be adapted to the moregeneral setting of simple knots in lens spaces. Indeed, the combinatorics of grid diagramsfor knots in lens spaces is completely analogous to that of knots in the three-sphere [3]. References [1] K.L. Baker. Small genus knots in lens spaces have small bridge number. to appear
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