aa r X i v : . [ m a t h . G T ] F e b ON KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS
LAWRENCE ROBERTS Introduction
Let L be a link in A × I where A is an annulus. We consider A × I to be embedded in R × R respecting the obvious fibration and embedding A into a round annulus in R . We alwaysproject L into R (or A ) along the R -fibration. The complement of L in A × I is therebyidentified with the complement of B ∪ L in S where B an unknot as depicted below, calledthe axis of L . We assume throughout that L intersects the spanning disc of B in an oddnumber of points. For example, B L
Let Σ( L ) be the branched double cover of S over L , and let e B be the pre-image of B inΣ( L ). Then e B is a null-homologous knot in Σ( L ) and we can try to compute \ HF K (Σ( L ) , e B, i ) = M { s | h c ( s ) , [ F ] i = 2 i } \ HF K (Σ( L ) , e B, s )where s is a relative Spin c structure for e B and [ F ] is the homology class of a pre-image ofa spanning disc for B . A particularly interesting case will be when L is a braid. Then thepre-image of the open book of discs with binding B is an open book with binding e B .To obtain a clean statement we need to adjust L by adding two copies of the center of A which are split from the remainder of L . We call this new link L ′ . The effect on thebranched double cover is to produce Σ( L ) S × S containing a knot e B B (0 , Proposition 1.1.
Let L be a link in A × I ⊂ R × R as above. Let L ′ be the adjusted versionof L . There is a spectral sequence whose E term is isomorphic to the reduced Khovanovskein homology of the mirror, L ′ , in A × I with coefficients in F and which converges to ⊕ i ∈ Z \ HF K (Σ( L ) ( S × S ) , e B B (0 , , i, F ) . The author was supported in part by NSF grant DMS-0353717 (RTG).
In [12] P. Ozsv´ath and Z. Szab´o constructed a spectral sequence which converged to d HF ( Y )for Y a double branched cover of a link in S . This spectral sequence featured the reducedKhovanov homology of the mirror of the link as the E term. The previous proposition isa generalization of this result.In the first half of this paper, we review the skein homology, first constructed in [1], andexamine its relationship to Khovanov homology. We then describe a spanning tree approachto computing this homology theory. This complex allows us to analyze the situation of L being alternating. Once this is completed we turn to building the relationship with knotFloer homology.In the second half, we derive the relationship between the two theories as the spectralsequence explained abover. We then turn to deriving some consequences of these spectralsequences. First, in [13], O. Plamenevskaya constructed a special element, e ψ ( L ), of theKhovanov homology of a braid and showed that it is an invariant of the transverse isotopyclass of the braid. She suggested that for certain knots, should this element survive in thespectral sequence, it would yield the contact invariant of the contact structure lifted from S to the double branched cover branched over the transverse knot. This element is also aclosed element in the skein homology where it defines the unique minimal filtration level.From these considerations we can prove Proposition 1.2.
Suppose there exists a n such that (1) ψ ( L ) is exact in the reduced Khovanov homology (2) The link surgery induced spectral sequence on
X/X − g collapses at E .then c ( ξ ) = 0 . where c ( ξ ) is the contact element for the lifted open book. The notation in this propositionis explained in section 8.Furthermore, for L alternating for the projection A × I → A much more can be said.We use the analysis of the skein homology for alternating L to prove the main theorem ofthe paper, theorem 9.1. Theorem 1.3.
Let L be a non-split alternating link in A × I intersecting the spanning discfor B in an odd number of points. Then for each k there is an isomorphism \ HF K ( − Σ( L ) (cid:0) S × S (cid:1) , e B B (0 , , k ) ∼ = M i,j ∈ Z H i ; j, k ( L ) where, for each Spin c structure, the elements on the right side all have the same absolute Z / Z -grading. Together these isomorphisms induce a filtered quasi-isomorphism from the E -page of the knot Floer homology spectral sequence to that of the skein homology spectralsequence. Thus the knot Floer spectral sequence collapses after two steps. Furthermore, forany s ∈ Spin c (Σ( L )) we have that τ ( e B, s ) = 0 where e B is considered in Σ( L ) . N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 3
In a sequel to this paper, we reprove the above theorem for Z -coefficients and use it toanalyze a class of fibered knots in certain three manifolds. Acknowledgements:
The author would like to thank John Baldwin and Olga Plamenevskayafor some very useful correspondence.2.
The Reduced Khovanov Skein Homology of [1]Throughout we will assume all coefficients are in F and suppress the ring notation. Thissection gives a brief description of a reduced form of the theory in [1] for categorifiying theKauffman bracket skein module for the I -bundle A × I and its relationship with the reducedKhovanov homology. We adjust the account in [1] to conform to that of Bar-Natan, [3].This alters the gradings from [1] to more directly related to Khovanov’s original definition.Pick an order for the c ( L ) crossings in a projection, P , of L to A . Let R be an ele-ment of { , } c ( L ) , then associate to R a collection of disjoint, simple, unoriented circles in A by resolving the crossings of P according to: −→ −→ We denote the resulting diagram by P ( R ). Let I ( R ) be I ( R ) = X m i where R = { m , . . . , m n } Finally, call an unoriented circle resulting from the resolution trivial if it bounds a disc in A , and non-trivial if it does not.An enhanced Kauffman state is then a choice of resolutions, R , and a choice of { + , −} for each of the resulting circles. As usual the enhanced states will be the generators of thechain groups. We define two bi-graded modules V ∼ = F v + ⊕ F v − and W ∼ = F w + ⊕ F w − where deg( v + ) = (1 , w + ) = (1 ,
0) and deg( v − ) = − deg( v + ), deg( w − ) = − deg( w + ).If the resolution R results in m trivial circles and l non-trivial circles we associate to R thebi-graded module V R ( L ) = V ⊗ l ⊗ W ⊗ m { ( I ( R ) , } We will refer to the first grading in the ordered pair as the q -grading and the second as the f -grading.The r th chain group, C r is then ⊕ { R | I ( R )= r } V R ( L ). These will form the components ofa complex, C , and the Khovanov skein complex will be C [ − n − ] { ( n + − n − , } for someorientation on the link L . The shift in [ · ] occurs in the dimension of the chain group. Thislast set of shifts will be called the final shifts. We will often only be interested in relativegradings, and so will sometimes ignore the final shifts. The complex before the final shifts We follow Bar-Natan’s shifting conventions
L. ROBERTS will be called unshifted .We now define the differential in the complex. As usual, we specify what happens when twocircles merge in a 0 → w + ⊗ w + → w + v + ⊗ v + → w + ⊗ w − , w − ⊗ w + → w − v + ⊗ v − , v − ⊗ v + → w − w − ⊗ w − → v − ⊗ v − → v ± ⊗ w − , w − ⊗ v ± → w + ⊗ v ± , v ± ⊗ w + → v ± The relevant maps for dividing are w − → w − ⊗ w − v + → v + ⊗ w − v − → v − ⊗ w − w + → w − ⊗ w + + w + ⊗ w − w + → v + ⊗ v − + v − ⊗ v + where the rule for w − is determined by the topological type of the circles in the result (twotrivial or two non-trivial circles). Theorem 2.1. [1]
The tri-graded homology, H ( L ) , of the complex C [ − n − ] { ( n + − n − , } with the differential defined above is an invariant of the oriented link L in A × I . Proof:
Let S ( P ) be the set of enhanced states and define for S ∈ S ( P ) τ ( S ) = { positive trivial circles } − { negative trivial circles } Ψ( S ) = { positive non − trivial circles } − { negative non − trivial circles } J ( S ) = I ( S ) + τ ( S ) + Ψ( S )Let S ijk ( P ) be the subset of S ( P ) with I ( S ) = i, J ( S ) = j, and Ψ( S ) = k . Define C i ; jk ( P )to be the free abelian group generated by S ijk ( P ). It is shown in [1] that the maps abovedefine a differential on C i ; jk ( P ) which increases the i grading by 1. Actually, this is provedwith J ′ ( S ) = I ( S ) + τ ( S ), but as the differential does not change k , the proof applies hereas well. Their homology is RII and
RIII invariant. With the shifts from a choice of ori-entation on the link, the theory we have outlined is also RI invariant. As with translationfrom Viro’s notation to Bar-Natan’s the shifts at the end are also necessary to pin downan invariant grading for RII, but the relative graded theory is invariant regardless. ♦ Let B ( A ) ∼ = { , , . . . , } be the set of all link diagrams in A with no crossings or trivialcomponents. Using the rules= + tq , L ∪ (cid:13) = ( q + q − ) L. N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 5 we can associate an element of Z [ q ± , t, B ( A )] to any diagram of L , denoted [ L ]. If we mapthe monoid B ( A ) to Z [ q ± , x ± ] by 1 → qx + q − x − we get a map φ : Z [ q ± , t, B ( A )] → Z [ q ± , t, x ± ]. After orienting L , let V ( t, q, x ) = t n − q n + − n − φ ( L ) which equals P k ∈ Z q k, L x k where q k, L = χ q,t ( H ∗ ; ∗ k ( L )) = X i,j t i q j rk F (cid:0) H i ; jk ( L ) (cid:1) The Euler characteristic for the skein homology is then V ( − , q, x ) and is an isotopy in-variant of L in A × I . On the other hand V ( − , q,
1) is the Jones polynomial as describedby Khovanov (see also [3]).There is also a reduced version of this theory. We mark the circle in P that is closestto the center, at the point intersecting the spanning disc for B . Every diagram P ( R ) inher-its this marking. Note that the marked circle in the resolved diagrams may be either trivialor non-trivial. The reduced homology is then defined to be the homology of the quotientof the above complex by the subcomplex generated by the enhanced states assigning a − sign to the marked circle. The reduced chain groups are denoted e V L ( P ) and the overallhomology by e H i ; jk . Lemma 1.
For each j , there is a spectral sequence whose E term is ⊕ i,k H i ; jk ( L ) andwhich collapses at E to ⊕ i H i,j ( L ) where H i,j ( L ) is the usual Khovanov homology for theembedding L → A × I → S . This statement also applies to the reduced theory. Proof:
The entire construction has been performed so that by ignoring the distinctionbetween trivial and non-trivial circles we obtain the Khovanov chain groups, i.e. if we use L → A × I → R × I as an embedding of L in S and ignore the axis. In this case we neglectthe f -grading and treat v ± and w ± the same. The maps defining the differential above arealmost those for the Khovanov homology, with the exception of a few terms which havebeen dropped. These terms are boxed below: v + → v + ⊗ w − + v − ⊗ w + v + ⊗ v + → w + v + ⊗ w − , w − ⊗ v + → v − w − → v − ⊗ v − Each of these terms preserves the q -grading, increases the i grading by 1, but decreases the f -grading by 2. Thus, the axis can be seen as filtering the Khovanov homology, with the E term of the corresponding spectral sequence being the Khovanov skein homology. Sincethe maps in the spectral sequence also preserve the − subcomplex, this conclusion occursfor the reduced homology as well. ♦ Lemma 2.
Let L be the mirror of L . Then there is an isomorphism H i ; jk ( L ) ∼ = H − i ; − j, − k ( L ) L. ROBERTS where H i ; jk is the corresponding cohomology group. Over a field, F , the last group is alsoisomorphic to H − i ; − j, − k F ( L ) . Furthermore, the spectral sequence converging to Khovanovhomology on H ∗ ; ∗∗ ( L ) is filtered chain isomorphic to that induced on the cohomology groups H ∗ ; ∗∗ ( L ) by the higher differentials on H ∗ ; ∗∗ ( L ) . Proof:
Each state for L defines a state for L by reversing the sign assignment on eachcircle. In addition, 0 resolutions are now 1 resolutions and vice-versa. Thus, i → c ( L ) − i , j → c ( L ) − j , and k → − k in the unshifted theory. Examining the differential for betweentwo states shows that the differential for L is the differential for the cohomology of L .Furthermore, after the final shifts we have ( i, j, k ) → ( i − n − , j + n + − n − , k ) for L and( c − i, c − j, − k ) → ( c − i − n + , c − j + n − − n + , − k ), where n − and n + refer to L . Thislast triple equals ( − ( i − n − ) , − ( j + n + − n − ) , − k ). For coefficients in a field standardhomological algebra implies that H F i ; jk ( L ) ∼ = H i ; jk F ( L )Carefully examining the terms giving rise to the spectral sequence shows that these mapto the terms in the spectral sequence on the cohomology. ♦ Since the k -grading filters the Khovanov complex, we can define for any element ξ ∈ KH i,j ( L ) a number T L ( ξ ) = min { k : ξ ∈ Im (cid:0) H ∗ ( ⊕ l ≤ k C i ; jl ) → KH i,j ( L ) (cid:1) } . When L is an unknot these numbers satisfy a relation similar to the τ invariant in knotFloer homology. Lemma 3.
Assume L is an unknot and let L be its mirror image. Let u ± be the generatorsof the Khovanov homology of the unknot in q -gradings ± . Then T L ( u ± ) = − T L ( u ∓ ) Proof:
Let F j ; s = ⊕ i ; k ≤ s C i ; jk ( L ) and let C j = ⊕ i,k C i ; jk ( L ). Since the differential preservesthe q -grading, j , there is a long exact sequence:0 −→ F j ; s I s −→ C j P s −→ Q j ; s −→ Q j ; s is the quotient complex, C j / F j ; s . Now ⊕ j H ∗ ( C j ) = Z u + ⊕ Z u − , and T L mea-sures the first s for which the map in the long exact sequence on homology will include u ± in the image of I s ∗ relative to the q -grading.There is a duality isomorphism D : H i ; j ( U ) → H − i ; − j ( U ), D ( u ± ) = u ∓ , on the Kho-vanov homologies which is induced by the symmetric pairing a + ⊗ a − m −→ a − ǫ −→ ǫ : A → Z is the counit for the Frobenius algebra underlying Khovanov homology. Inparticular, a + → h a + , ·i = a ∗− . This can be extended to V as well, and corresponds tochanging the markers on each of the circles in an enhanced state. It thus induces a map on N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 7 the skein homology spectral sequences. Checking the effect on the differential establishesthe following commutative square:00 Q ∗− − s − ( U ) C ∗− ( U ) F +1 ,s ( U ) C ( U ) F ∗− , − s − ( U ) Q +1; s ( U ) 00 ....................................................... .......................................................................................... ............ ....................................................................................................... ............ P ∗− s − ....................................................................................................................................... ............ I s ............................................................................................................................. D ............................................................................................................................. D ....................................................................................................... ............ I ∗− s − ....................................................................................................................................... ............ P s ............................................................................................................................. D .............................................................................. ................................................................... ............ If u + is in the image of I s , then u ∗− is in the image of P ∗− s − . In paticular, I ∗− s − ( u ∗− ) = 0.But then there is no element in F − , − s − which maps to u − and so − s − < T L ( u − ).Thus if s = T L ( u + ) then − T L ( u + ) ≤ T L ( u − ). If u + is not in the image of I s – i.e. s < T L ( u + ) – then I ∗− s − ( u ∗− ) = 0. Choose some element on which this image pairs non-trivially and is uniformly in q -grading −
1. This element must then map in homology to a − and − s − ≥ T L ( u − ). In particular, if s = T L ( u + ) − − T L ( u + ) ≥ T L ( u − ). Thisproves the result. ♦ Let L and L be two links in A × I . Let L = L | L be the link in A × I where A = { z : 1 ≤ | z | ≤ } and L lies in { z : 1 ≤ | z | ≤ }× I while L lies in { z : 2 ≤ | z | ≤ }× I .Then we can prove Lemma 4.
With coefficients in a field, F , there is an isomorphism H i ; jk ( L ) ∼ = M i + i = i, j + j = jk + k = k H i ; j k ( L ) ⊗ H i ; j k ( L ) In fact, this is an isomorphism of spectral sequences, so that if ξ ∈ KH i ; j ( L ) and ξ ∈ KH i ; j ( L ) then T L ( ξ ⊗ ξ ) = T L ( ξ ) + T L ( ξ )Finally, if there is a non-trivial component, L , split from the rest of L , it can be made tolie in the diagram without crossing any other strand of L . L survives unchanged in everyresolution; thus, marking it induces a marking on a non-trivial circle for every resolutionWhen the number of intersections of L with the spanning disc for B is odd, then the reducedskein homology of this configuration has the form e H ( L − L ) ⊗ V . This choice shifts thecomplex by { (1 , } , so we will always shift at the end to compensate. Thus, the final shiftwill be [ − n − ] { ( n + − n − − , − } for this marking convention. This component is specialand doesn’t participate in the calculations. Thus for the mirror, we keep it labelled + andderive the same duality for the mirror image as previously.3. Spanning Tree Complex
As with Khovanov homology, the skein homology for links in A × I with connected projec-tions admits another presentation in terms of the spanning trees for the knot diagram. We L. ROBERTS follow [16] in establishing this result, but see also [4].Start with the projection of L in the plane and follow Wehrli’s algorithm. First, num-ber the crossings. Now proceed to resolve the first crossing if both resolutions produceconnected diagrams. The homology can be shown to be a mapping cone on these resolu-tions. If resolving the first crossing disconnects the diagram for one or other resolution,proceed to the next crossing until you come to one where both resolved diagrams are con-nected or you run out of crossings. Now iterate this procedure on the resolved diagrams.The result is a tree of diagrams, the resolution tree, whose leaves correspond to unknotsthat are reducible to the standard unknot using only the first Reidemeister move. There isa unique way to smooth the remaining crossings to get an unknot in the plane. Likewiseto each complete smoothing which produces an unknot in the plane there is a unique leafwhich smooths to it (due to the enumeration of the crossings). Let K ( L ) be the completesmoothings with only one component and for each S ∈ K ( D ) let D S be the twisted unknotcorresponding to S .We can prove that the unshifted skein complexes behave in the following way due to an RImove: C ( ) ∼ = C ( ) { ( − , } ⊕ B C ( ) ∼ = C ( )[1] { (2 , } ⊕ B where B and B are contractible. It does not matter whether the RI move involves trivialor non-trivial components in the complete smoothings. The shifts result from the invarianceof the theory after the final shifts are performed. For the first RI move above, the left sidewould need to be shifted [0] { (1 , } further than the right side, due to the extra positivecrossing. Thus the right side should be shifted { ( − , } to correspond to the left side inthe unshifted complex.With this observation we may proceed analogous to [16] to obtain the following propo-sition: Lemma 5. [16]
Let L ⊂ A × I have a connected diagram in A . Then there is a decomposition C ∼ = A ⊕ B where C is the unshifted version of the skein complex, B is contractible, and A is given by A = M S ∈ K ( L ) H ∗ ; ∗∗ ( D S )[ − w ( D S )] { ( − w ( D S ) , } [ r ( S )] { ( r ( S ) , } where w ( D s ) is the writhe of D s , and r ( D, S ) is the number of smoothings necessary inresolving L to get S . Of course, the unshifted homology H ∗ ; ∗∗ ( D S ) does not change while using ReidemeisterI moves, but the difference between considering D S in A and in the plane is precisely todisallow RI moves which would need to cross B . Thus D S can be simplified to D ′ S , a twistedunknot, where all the twisting ultimately must link with B . This implies that D ′ S is isotopicto a knot of the special form in Figure 1, where each n i records the number of half twists. N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 9 n − n ± n k ∓ n k − Figure 1.
The special class of unknots which act as the base cases for thespanning tree complex.The homologies of these knots form the building blocks of the spanning tree complex above.Since we assumed that the number of intersections with the spanning disk is odd, the dia-gram D S must be non-trivial, and thus link B . Suppose a connected diagram, fully reducedin A × I does not look as above. Choose a region “under” B and follow it around clock-wise and counter-clockwise. Suppose in both directions we come to crossing regions whichtraverse the region, as we must since the diagram is connected. Suppose that these twistregions are distinct. Since the diagram is fully RI reducible, only the strands of one of thetwist regions can link B . The other must then reduce using RI moves, since the diagram isessentially planar. This forces a knot isotopic to one of the unknots seen above.We can say a little more concerning the unknots in 1. In particular, we compute thenumbers, T L , for these unknots. Let T ± ) be the number of left/right-handed twist re-gions in Figure 1. Since these are unknots, their Khovanov homologies are composed of F u + in homological and q -grading (0 ,
1) and F u − in (0 , − L linksthe axis an odd number of times. We can then prove: Proposition 3.1.
For the special unknotted branch loci in Figure 1, let T ( L ) denote T − ) − T + ) . Then T L ( u ± ) = T ( L ) ± . For the alternating unknots in this family, T L ( u ± ) = ± . Proof:
These unknots are isotopic to the standard planar unknot using only RI -moves.M. Jacobsson provides rules for mapping closed elements in the Khovanov cube of a link tothose of the link with a single RI move, which in our notation are as in Figure 2. We canuse these moves to try to compute T L ( u ± ). As a first step, we exhibit a specific generator ∈ − + ++ + − − −− − ∈ ++ − − + Figure 2.
Rules for transfering generators when an RI move is applied.The particular twist is represented on the far right. Note that these mapsare chain maps inducing isomorphisms on the Khovanov homologies, [7]
Figure 3 which will produce u ± in homology. The maximal value of k needed to obtain this gener-ator in ⊕ l ≤ k C i ; jk will then be an upper bound on T L ( u ± ).Consider an unknot as in Figure 3 formed by resolving all the crossings of L horizontally.We begin by examining the effect of replacing the outermost resolution with a crossing, N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 11 according to the Jacobbson rules. According to these rules, for left handed twist regions, a − on the outer circle will propagate to each new circle as we change the resolution at eachcrossing. Meanwhile a + maps to a linear combination of the two generators formed by a+ on one of the new circles and a − on the other, or vice-versa. For right handed twistregions the argument is slightly different. Namely, + markers are placed on the inner circlesregardless of the marker on the outer circle. This is a chain map since the disconnecteddiagram occurs as the 1-resolution for the new crossing.When resolving the outermost crossing there are four cases to consider:(1) The crossing occurs in a left handed twist region, and the original circle is labelledwith a − . Then each new circle will be labelled with a minus until we come to a righthanded twist region. All the non-trivial circles formed by the resolution changeswill be labelled with a minus and thus we have a contribution of − T + ) − k -grading, as this is the number of non-trivial circles in this group. The righthanded twist region which may follow will be of the type (2), to which we turn now.(2) The crossing is in a right handed twist region, and the circle is labelled with a − .Then all the new circles will be labelled with +’s until the next left handed region,which is of type (3). There are T − ) nontrivial circles which receive a + markingin this way. If the − is the outermost in the whole diagram, we have a contributionof T − ) −
1. If it comes from a left handed region preceding, say from case (1),then we have added T − ) to the amount already there, which we assume satisfiesthe proposition. Thus, we will still satisfy the proposition after this right handedregion, especially if it is the last. For example, if we pass from type (1) to type (2),we can associate each marker with the twist region on its right, including here theunbounded complement of the diagram as a left handed region. This gives a totalof T − ) − T + ) − − on the new circle, then all additional circles until we change handedness will have a − . However, if we place the + on the new circle, and a − on the old circle, we willgenerate a string of − ’s to the right of the +. If the + marker is on a non-trivialcircle, the total number of non-trival plus circles does not change, whereas therewill be T + ) minus markers introduced. If it is on a trivial circle, then there are T + )+1 minus markers introduced and these generators are in a smaller k -grading.Whatever marker winds up on the non-trivial circle at the junction with the nextright handed twist region is immaterial as both type (2) and (4) will propagate thesame number of additional + markers.(4) If the crossing is in a right handed twist region, and the circle is labelled with aplus we obtain plus markers on all the new circles until the next left handed twistregion. If the circle is the outermost, this is a contribution of T − ) + 1. If it follow another region, we have added T − ) to the running total, and the proposition isstill satisfied.Note that type (1) can only occur in the very outermost twist region, since the right handedtwist regions always pass a + marker to the next left handed region. It is type (3) whichtruly determines the outcome. Checking the numerics shows that the maximal k -gradingfor the generators in the linear combination so produced is T − ) − T + ) ± ± − marker on the unknot in Figure3. Since these are generators of the Khovanov homology for the unknot, and the maps inFigure 2 are the chain maps used to show the RI-invariance of Khovanov homology, wehave exhibited an element of the chain complex of the skein homology which survives thespectral sequence and will represent one of u ± depending on the original marker.Altogether, this shows that T L ( u ± ) ≤ T − ) − T + ) ±
1. However, the argument alsoapplies to L and we know that T L ( u ∓ ) = − T L ( u ± ). In the mirror image there are T + )left handed regions and T − ) right handed regions. Hence, T L ( u ∓ ) ≤ T + ) − T − ) ∓ − T L ( u ± ) gives T L ( u ± ) ≥ T − ) − T + ) ±
1, and the re-sult follows The final statement is simply a reflection of the even number of twist regions,alternating between handedness, when there are an odd number of strands. ♦ Results for the skein homology of alternating links
The goal of this section is to use the spanning tree presentation of the skein homology toprove the following theorem
Theorem 4.1.
Let L be an alternating link in A × I intersecting the spanning disc for B in an odd number of points. Then the Khovanov skein homology H i ; jk ( L ) is trivial unless k − j + 2 i = σ ( L ) . Thus the homology is determined by the Euler generating polynomial V ( t, q, x ) = t n − q n + − n − φ ([ L ]) , defined in section 1, and the signature of the oriented link σ ( L ) , thought of as embedded in S . We will follow [16] in calculating the Khovanov-type homology of an alternating configura-tion. Both provide simplified proofs of E. S. Lee’s result concerning alternating links, [8],which describes the result of computing the spectral sequence for the axis filtration: thehomology will be supported on the lines j − i = − σ ( L ) ±
1. It is towards a variation ofthis result that we now aim. Note, however, that our result is not just about supports. Wereturn to this at the end of the proof.Assume that L admits an alternating projection to A which is connected as a subset of A . We maintain the assumption that L intersects the spanning disc for B in an odd num-ber of points; however, we will relax this when it is to our advantage. We will bi-color the plane according to the following convention:For any L , regardless of the parity of intersecting the spanning disk, we define M ( L ) to bethe number N W − N B where N W is the number of white regions intersecting the projection N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 13 of B and N B is the number of black regions. When L intersects the spanning disc in anodd number of points, M ( L ) = 0; for an even number of points M ( L ) = ±
1. This numberdoes not change under Reidemeister moves applied to L , nor does it change when crossingsof L are resolved. Furthermore, all the projections, D S , in the spanning tree complex willbe alternating. We start with a lemma concering these unknots Lemma 6.
For each alternating twisted unknot in Figure 1 the homology H i ; jk ( L ) satisfies k − j + 2 i = M ( L ) . We will show that diagrams of the special form above have the property that H i ; jk satisfies k − j + 2 i = M ( D S ), and that the last number is determined by the type of crossing on theoutermost boundary. We start with the following cases:(1) L as a single non-trivial unknot has this property. Its homology is 0 unless ( i ; j, k ) = ± (0; 1 , F . But then k − j + 2 i = 0 = M since there isone black and one white region.(2) If D S has the property that k − j + 2 i = C so does D S ∪ N where N is a disjointnon-trivial circle.(3) The closures of σ ∈ B and σ − ∈ B have the property that k − j + 2 i = M ( D ).This requires a computation. For σ − the shifted complex has homology H i ; jk ∼ = (cid:26) F − ( j, k ) = ( − , F ( j, k ) = ( − , − , ( − , , (1 , i , and each element has k − j + 2 i = +1. Furthermore, N W = 2 and N B = 1, so M ( D ) = 1. For the closure of σ we obtain: H i ; jk ∼ = (cid:26) F ( j, k ) = (3 , F ( j, k ) = ( − , − , (1 , , (3 , k − j + 2 i = − M ( D ).The nontrivial unknot, and the closures of σ and σ − , are the base cases for our induc-tion. We now assume that we have a twisted unknot of the special type above, which has σ ( L ) = 0. See Figure 4 and Figure 5 to clarify the notation. We start by assuming thatnear the inner point where B crosses the plane the twisting is right-handed. Assume thatthere are n > I − n . If we 0 resolve the innermostcrossing we obtain I − n − ,while if we 1 resolve the crossing we obtain N ∪ I + . Let [ s ] { ( t, } be the contributionto the final shift of I − n arising from the crossings not involoved in this twist region. Thereis then a long exact sequence −→ H ∗ ( I + ∪ N )[ − s ] { ( − t, } [ n − { (2 n − , } [1] { (1 , } −→ H ∗ ( I − n )[ − s + n ] { ( − t + 2 n, } −→ H ∗ ( I − n − )[ − s + n − { ( − t + 2 n − , } −→ where the sequence arises from 0 → H ∗ ( I + ∪ N ) come from its arising in the 1 resolution and from the ad-ditional negative crossings introduced from the twists needed before the resolution change. B n crossings I − n B n − I − n − B N B N I + Figure 4.
A depiction of I − n for n >
1, and the corresponding I + , as itoccurs in the resolution tree for the innermost crossing.Those for H ∗ ( I − n − ) come from the negative crossings remaining in the resolved diagram.The maps two internal maps are degree preserving. If k − j + 2 i = C for the shiftedcomplex for I − n then k − j + 2 i = C + t − n + 2 − s + 2 n − C + t − s . Reversingthe shift for H ∗ ( I − n ) we obtain that elements mapping to the homology for I − n − satisfy k − j + 2 i = C + t − s − n + t + 2 n + 2 s = C . By assumption, C = M ( I − n − ) and M ( I − n ) = M ( I − n − ) since there has been no change in the black/white region count. Onthe other hand, M ( I + ) = M ( I − n ) − N does not change k − j + 2 i , if k − j + 2 i = M ( I + ) for I + , we see that the terms in the unshifted complex for I + satisfy k − j +2 i = M ( I + )+ t − − n +2+2( − s +1+ n −
1) = M ( I + )+ t − s +1. Applying the shiftsto get the shifted complex for I − n prodeuces elements with k − j + 2 i = M ( I + ) + 1 = M ( I − n ).Every element in the image of H ∗ ( I + ∪ N ) will also have the desired property. Thus byinduction, the property will be true also for I − n .This leaves the case where n = 1. The 1 resolution occurs in the same way and wemay draw the same conclusion. However, for the 0 resolution a large collapse can occur. If I + has m ≥ J − . The complex for J − is thus shifted by { ( − m, } wheninjected into that for I − . This implies that k − j + 2 i increases by m in the unshifted N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 15 I − B B I − B I + m N m crossings B J − Figure 5.
A depiction of I − , I + and J − , as they occur in the resolutiontree for the innermost crossingcomplexes. In the shifted complexes, I − is shifted [ − { ( m − , } more than J − . Thatshift reduces k − j + 2 i by 2 − m − − m . Thus after the final shift there is a difference of0. But note that the resolution eliminates both a black and a white region and thus leaves M ( J − ) = M ( I − ). All told, if k − j + 2 i = M holds for the knots with fewer crossings andthe innermost crossing is negative then it also holds for I − n .A similar argument can be deployed for the case where the innermost crossing is posi-tive. Alternately we can appeal to the symmetry under reflection to switch the two cases.Since this switches the black and white regions, it also multiplies M by − ♦ Thus for every unknot in our collection we have k − j + 2 i = M ( K ) for every generator inthe homology. In particular, ( j, k ) determines i . Note that this conclusion remains valid ifwe add a single marked non-trivial circle. It also remains true if we shift by [ − w ] {− w } . Aswith the original proofs of the alternating links property, the value of r ( S ) is the same forevery complete smoothing in K ( L ), depending only on the number of black regions and thecrossings joining them. So all the generators for the spanning tree model of the unshiftedhomology satisfy k − j + 2 i = r ( S ) after the [ r ( S )] { ( r ( S ) , } shifts and the odd number of intersections. The final shift of the diagram for L is [ − n − ] { ( n + − n − , } and produces gen-erators satisfying k − j +2 i = r ( S ) − n + . From [8], we have that r ( S ) − n + = σ ( L ). Thus, afterthe final shifting, every generator in the spanning tree complex satisfies k − j + 2 i = σ ( L )and since the differential in the spanning tree complex also preserves ( j, k ) and increases i , this is also the homology. For those generators which survive the spectral sequence tothe Khovanov homology, we also have that j − i = − σ ( L ) ±
1. Thus, for these generators, k = ± A comment about supports:
Wehrli’s argument produces an unshifted chain com-plex which has the same chain groups for l + r = i and 2 l + r ± j where r = r ( S ) isconstant. Thus j − i = − r ± j − i = − σ ( L ) ±
1. For a given q -grading, j , there are two i gradings differing by 1. Thus there can still be non-zero termsin the differential, which may result in torsion or vanishing homology groups, and thus thehomology is at most supported on these lines. In our case, these groups are distinguishedby their k -value, which is also preserved by the differential. The issue of torsion returns inthe spectral sequence, but it is known that at most 2 r -torsion occurs for alternating knots,[15], and so working over F will correct it.5. Resolutions in knot Floer homology
We now leave the Khovanov skein homology to establish some results linking it to knotFloer homology. The two will intertwine in later sections.Assume that L intersects a spanning disc for B generically in an odd number of points.Let R be a complete resolution of the crossings in P . Of the closed curves in P ( R ), somenumber, m , are geometrically split from the axis, B . The remainder, l , form an unlinkeach of whose components link the axis one time. For such a link of unknots, the doublebranched cover is easily computed to be l + m − S × S . Moreover, e B ( R ) is still a knotsince each unknot which is split from B intersects a disc generically an even number oftimes. This knot is l − B (0 , ⊂ l + m − S × S (and the unknot in m S × S if l = 1)where B (0 , ⊂ S × S S × S is the knot obtained by performing 0 surgery on any twoof the three components of the Borromean rings. Hence, \ HF K ( e B ) ∼ = V ⊗ ( l − ⊗ W ⊗ m where V ∼ = Z ( , ) ⊕ Z ( − , − ) and W ∼ = Z ( , ⊕ Z ( − , . Here the first term in the sub-script is the rational grading, whereas the second term is the filtration. Since l − Spin c structure. Note that there are no higher differentials in theknot Floer spectral sequence. All we have done is compartmentalize in a new manner theHeegaard-Floer homology of l + m − S × S . It is the latter which is associated to R in [12].We wish to define an isomorphismΦ B ( R ) : e V L ( R ) ∼ = −→ \ HF K (Σ( P ( R )) , e B ) N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 17 for any complete resolution R . However, a slight mismatch arises: the knot Floer homologyof the binding implicitly corresponds to marking a non-trivial circle. This cannot alwaysbe arranged in the skein homology theory. We rely upon a trick to resolve this problem:we introduce two non-trivial circles into L which link B once and otherwise do not interactwith the diagram. These should be considered innermost circles. We always mark theinnermost one (we need two to keep the binding connected) and since this circle does notinclude any crossings, it will be the marked circle throughout.The effect on the double cover of changing L to L ′ is to replace Σ( L ) with Σ( L ) S × S and to replace e B with e B B (0 , B and then examining the double cover of a small ball which includesthem and an arc on B . The effect of these connect sums on the Heegaard-Floer homology iswell understood. In particular, since B (0 ,
0) induces an entirely collapsed spectral sequencefor the Heegaard-Floer homology, we will be able to read off any information about the knotFloer homology of e B from that of e B B (0 , P ( R ) by the marked circle first, then all thenon-trivial circles, then all the trivial circles. An element of e V ( P ( R ) , B ) is encoded as+ ⊗ v ± ⊗ · · · ⊗ w n ± and is mapped to γ i · · · γ i k · Θ + where { i , . . . , i k } are the indices forthe minus signs on non-marked circles, γ j is the first homology class dual to the j th sphere,and Θ + is the highest degree generator of d HF (Σ( P ( R ))). In particular, a representativefor γ j in F -homology can be found by lifting an arc between the marked circle and the j th circle. 6. Filtering section 6 of [12]Next we associate maps in the knot Floer homology to the changes in resolutions at cross-ings. In our case, these maps become maps between filtered groups. We work backwardsfrom the maps in [12].First, we note that the resolution changes occur in three ways: between circles split fromthe axis, between circles linking the axis, and between circles of mixed linking. The firstoccur precisely as in [12] due to the local nature of the surgeries in the double cover andthe connected sum decomposition of the covering manifolds. In particular, the maps for thefiltered theory are just the maps for the unfiltered theory tensored with the identity on thetensor products of the V -vector spaces. Hence they reflect the differential of the reducedKhovanov homology.Now consider a resolution change joining two circles which link the axis. In the doublecover, this corresponds to a cobordism which involves 0-surgery on a curve which is homo-logically non-trivial and intersects only those spheres intersecting the binding. Such a cicleis isotopic to a circle in a fiber of the open book determined by B (0 ,
0) before connectsumming with extra S × S ’s. Moreover, since the circle is the lift of an arc between two branch points, it is homologically non-trivial in the fiber. Ignoring the choice of basisimplicit in the above description, we can calculate the effect of such a surgery by looking atthe standard picture of B (0 ,
0) and doing 0 surgery on a meridian of one of the 0-surgeredcomponents of the Borromean rings. When we connect sum with copies of B (0 ,
0) we obtaina diffeomorphic picture to the one described above. We then use homology classes to pindown the maps in the original picture. In the unfiltered version, the model calculation usesthe following long exact sequence (which must split as depicted due to ranks and gradings). · · · F ⊕ F − F ⊕ F ⊕ F − F ⊕ F − · · · .................................................................................................................................................... ............ − ............................................................................................................ ............................................................................................................................. ................. − ............. ............. ............. ............. ............. ............ ............ ............................................................................................................................................................................. ............ − In the identification with Khovanov homology, the F term corresponds to v + ⊗ v + and itthus maps to w + . The term mapping to F − in the surjection is the image under ν of F where ν is the meridian we do not surger. Meanwhile the image of ν is annihilated.Transfering back to the basis from the resolution, this tells us that γ + γ generates thekernel, and γ and γ are mapped isomorphically to γ ′ . Transfering back further to Kho-vanov’s notation, we get γ → v − ⊗ v + → w − ← γ ′ and v + ⊗ v − → w − .For the filtered version, we obtain the model long exact sequence which filters the aboveone: · · · F − F ⊕ F F F ⊕ F − F − F ⊕ F F · · · ................................................................................... ............ ................................................................................................................................................................................................................................................................................................................................................................................................ .................................................................................................... ............ ................................................................................... ...................................................................... .................................................................................................... ............ The first term is the knot Floer homology of B (0 , S × S which we obtain after the 0-surgery on the meridian; the thirdterm is the result of +1-surgery on the meridian, B (0 , −
1) in the notation of [11]. Thegrading and ranks again determine the filtered maps on the first page. When we join twocurves which link the axis, we obtain one which does not link the axis. This can be seenby considering the possible winding numbers for the result: 0 or 2. However, the resultis a Jordan curve in the plane and thus cannot have winding number 2 about the origin.Working back through the basis transformations as before, these correspond in our notationto the maps v + ⊗ v + → v + ⊗ v − → w − , v − ⊗ v + → w − and v − ⊗ v − → N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 19 v v w ν f B The surgery circle, ν , annihilates γ + γ again in mod-2 homology. The result of theresolutioin change is now a circle which links the axis. The relevant cobordism map is from B (0 , S × S → B (0 ,
0) and corresponds to v + ⊗ w + → v + , v + ⊗ w − → v − ⊗ w + → v − and v − ⊗ w − →
0. This can be seen from the following graded exact sequence: · · · F − ⊕ F − F − ⊕ F F ⊕ F F − F F F − F F · · · ................................................................. ..................................................................................... ............................................................................................... ............ ............................................................ ............................................................................. .......................................................................................... .......................................................................................... .................................................................................................... .................................................................................................... ................. ............................................................................................................ ............................................................................................................................. ............................................................................................................................. ............ where F corresponds to v ⊗ v ⊗ w + and is mapped to v ⊗ v ′ + , taking into account both0-framed knots in the Borromean rings. Note that a w − always forces the map to be 0.Due to the introduction of the two new components we do not need to examine whathappens if one of the circles is the marked circle: a division or merging never includes themarked circle.Similar considerations, or duality, establish the maps for the case of splitting a circle intotwo circles. Note that the above maps are from +1 resolutions to 0 resolutions. This forceus to use the mirror of L in establishing the relationship between the knot Floer homologyof e B and the reduced skein homology. Proposition 6.1.
Let P be a projection for L ′ ∪ B . Let R be a choice of resolution foreach crossing of L ′ . Then there is an isomorphism Φ B ( R ) : e V ( P ( R ) , B ) ∼ = −→ \ HF K (Σ( P ( R )) , e B ) Let R ′ be a resolution found by changing a single smoothing in R from to +1 . Then thefollowing diagram commutes \ HF K (Σ( P ( R )) , e B ) \ HF K (Σ( P ( R ′ )) , e B ) e V ( P ( R ) , B ) e V ( P ( R ′ ) , B ) ............................................................................................................................. Φ B ( R ) ............................................................................................................................. Φ B ( R ′ ) ................................................................. ............ b F R Let L = L ∪ . . . ∪ L n be a framed link in a three manifold Y . Following section 4 of [12] welet R = ( m , . . . , m n ) where m i ∈ { , , ∞} and Y ( R ) be the result of f r ( L i )+ m i µ i surgeryon each L i where µ i is the meridian of L i and ∞ -surgery is µ i -surgery. We let 0 < < ∞ define a lexicographic ordering on { , , ∞} n and call I ′ and immediate successor to I asin [12]: all the m ′ j are the same as m j except for one where m ′ i > m i , excluding the case( m ′ i , m i ) = ( ∞ , I ′ of I there is a map F R Let L = L ∪ . . . ∪ L n be a framed link in ( Y, K ) such that lk( L s , K ) = 0 for all s . For each integer k , and surface F spanning K and disjoint from L , there is aspectral sequence such that (1) The E page is ⊕ R ∈{ , } n \ HF K ( Y ( R ) , K, k )(2) The d differential is obtained by adding all b F R Combine section 8 of [11] with section 4 of [12] to construct a complex X whichrecords the maps induced from the surgeries described above. Use the approach in section N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 21 L consists of a singlecomponent. For the induction in the proof of Theorem 4.1 of [12] we note that the adjustedstatement is that E ( X ) and E ( X ( S )) are isomorphic to 0, by applying the stated reasonsto the E maps. This allows us to repeat the induction and conclude that the the knotFloer complex is filtered quasi-isomorphic to an iterated mapping cone, namely the analogof the surgery complex built in [12], which will be denoted X ( L ). ♦ We consider the link in a connect sum of S × S ’s upon which we perform the surgeriescorresponding to the resolution changes. These are all algebraically split from the binding,so the above proposition applies for the choice of framing coming from the crossing data.Combining with the proposition from the previous section, and adjusting for the filtrationinformation, allows us to identify the E -page with the skein homology complex. As in [12]we may then conclude Proposition 7.2. Let L be a link in A × I ⊂ R × R as above. There is a spectral sequencewhose E term is isomorphic to the reduced Khovanov skein chain complex of L ′ in A × I withcoefficients in F and which converges to ⊕ k ∈ Z \ HF K (Σ( L ) ( S × S ) , e B B (0 , , k, F ) . By splitting according to the filtration data we can obtain the slightly stronger Proposition 7.3. There is a spectral sequence whose E term is isomorphic to the sub-complex of the reduced Khovanov skein complex of L ′ generated by the enhanced states with Ψ( S ) = 2 k and which converges to \ HF K (Σ( L ) ( S × S ) , e B B (0 , , k ) . In fact, by taking the direct sum of all the groups for the knot Floer homologies over all theresolutions we can obtain a bi-filtered complex, filtered by the pair ( I, Ψ), where the E term corresponds to the filtration of the bi-filtered reduced Khovanov homology complex.Using the graded objects for just the Ψ filtration and taking their homology produces thefirst proposition above. The additional terms in the maps in the Khovanov complex inducemaps in the E level of the spectral sequence using the Ψ filtration, since these correspondto terms in the filtered cobordism maps between the Heegaard-Floer homologies. Thesemaps fit together to provide a filtered version of the spectral sequence in section 4 of [12]with K inducing the filtration. Additional pages ultimately calculate the Heegaard-Floerhomology of the branched double cover.More can be concluded from the proof outlined above and the homological algebra in theappendix. Lemma 7. For each r ≥ , the E r page of the spectral sequence for d HF (Σ( L ) ( S × S )) computed from ⊕ k ∈ Z \ HF K (Σ( L ) ( S × S ) , e B B (0 , , k, F ) using the differential fromthe knot Floer homology is quasi-isomorphic to the E r page for the filtered complex X ( L ) ,computed using the maps induced from the link surgeries spectral sequence above. Thus, the values of Ψ will filter the Heegaard-Floer homology groups of the double branchedcover in a way corresponding to that induced by the spectral sequence for e B (namely, theassociated graded groups will be isomorphic).8. Transverse links, open books and contact invariants First, we note that Theorem 1. Any transverse link is transversely isotopic to a braid closure. Furthermore,two braids represent transversally isotopic links if an only if one can be obtained from theother by conjugations in the braid group, positive Markov moves, and their inverses. This is the culmination of work by Bennequin for the first part, and by V. Ginzburg, S.Orevkov, and N. Wrinkle, who independently proved the second part. We will replace thecontact structure with an open book. The standard contact structure on S is supported bythe open book with unknotted binding and discs for pages. In the braid picture, this cor-responds to including the axis of the braid, which is an unknot. When we take a branchedcover of a transverse link, the contact structure lifts to a contact structure in the coverwhere we use a Martinet contact neighborhood of the transverse link. In the open bookpicture, this contact structure is supported by the pre-image of the open book, whose fibersare now more complicated, but whose binding is the lift of the axis. This follows sincethe lifted contact structure remains C -close to the pages of the open book, and transverseto the binding. We call this contact structure ξ . The contact structure on (cid:0) S × S (cid:1) induced by the fibered knot B (0 , 0) will be denoted ξ .For a braid, O. Plamenevskaya, [13], [14], defines a cycle, e ψ ( L ), in the reduced Khovanovhomology chain group. First she resolves all the crossings in the direction of the orientedbraid. This constructs the maximal number of non-trivial loops in the skein algebra per-spective. She then labels every one of the unmarked strands with a − and the markedstrand with a +. This enhanced state is closed in the reduced Khovanov homology theory,[14].Let L be a braid whose closure is the transverse link. Theorem 8.1. Suppose L intersects the spanning disc for B an odd number of times.Then the element e ψ ( L ′ ) is closed in the skein Khovanov homology and represents the uniquehomology class with minimal Ψ -grading. Under the correspondence with the E term of thespectral sequence converging to knot Floer homology, it maps to an element which survivesthe spectral sequence and generates \ HF K ( − Σ( L ) (cid:0) S × S (cid:1) , e B B (0 , , − − g ( e B )) ∼ = F . Upon mapping this last group into d HF ( − Σ( L ) (cid:0) S × S (cid:1) ) , e ψ ( L ′ ) corresponds to thecontact element c ( ξ ξ ) . Note: The correspondence at the end is not the same as first mapping ψ into the reducedKhovanov homology and then considering the spectral sequence from it to the Heegaard-Floer homology of the branched double cover. Proof: There is only one element in the skein chain group which has Ψ = − g ( e B B (0 , N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 23 and that is Plamenevskaya’s element. For if b is the braid index of L ′ then Euler charac-teristic calculations imply that 1 − g ( e B B (0 , − b and thus Ψ must equal 1 − b .This can only happen when all the crossings are resolved in the direction of the link sothat there are b non-trivial circles and precisely one circle (the marked one) is adornedwith a + sign. e ψ ( L ′ ) is characterized as the unique enhanced state with minimal value forΨ( S ) and thus generates the homology in this f -grading. This enhanced state survives inthe spectral sequence for the knot Floer homology of the binding and yields in the limit agenerator of \ HF K ( − Σ( L ′ ) , e B B (0 , , − g ( e B B (0 , ∼ = F since it is the only generatorin the filtration level.The branched cover of B over L ′ is e B B (0 , 0) which supports the contact structure ξ ξ . The contact element c ( ξ ξ ) is the image in d HF ( − Σ( L ′ )) of the generator of \ HF K ( − Σ( L ′ ) , e B B (0 , , − g ( e B B (0 , − g ( e B B (0 , d HF ( − Σ( L ′ )). Thislevel is either ∼ = F or ∼ = 0 depending upon whether the contact element vanishes. Thus,Plamanevskaya’s element converges to the contact element in the Heegaard-Floer homology(with F coefficients). ♦ Corollary 1. Under the correspondence in the previous theorem, e ψ ( L ) corresponds to c ( ξ ) ∈ d HF ( − Σ( L ) , F ) . Proof: If L intersects the spanning disc for B an even number of times, use a positiveMarkov move to increase the number of strands by 1. e ψ ( L ) is mapped to e ψ ( L + ), [14],under this move. Meanwhile, in the double cover this corresponds to positively stabilizingthe open book, and thus does not change the contact invariant. Renaming L + by L we maynow assume L intersects the spanning disc an odd number of times. Furthermore, e ψ ( L )clearly corresponds to e ψ ( L ′ ) in a precise way. Using the previous theorem we have that e ψ ( L ′ ) maps to c ( ξ ξ ) under the spectral sequence and c ( ξ ξ ) = c ( ξ ) ⊗ c ( ξ ). Addingthe two meridional strands tensors both homologies with V ⊗ . Thus, c ( ξ ) in the knotFloer homology of e B corresponds to Plamenevskaya’s element in the skein homology of L since both are alterred in the same formal manner by the introduction of the new strands. ♦ We now turn to proving a the non-vanishing result mentioned in the introduction. Webegin with a lemma: Lemma 8. Let C be a bifiltered complex over a field. Then up to isomorphism there is aunique bifiltered complex C ′ such that (1) C ′ is bifiltered chain homotopy equivalent to C (2) C ′ ij ∼ = H ∗ ( C ij )(3) The differential d ′ = P d ′ ij on C ′ has d ′ = 0 , and induces the same spectral se-quences for both filtrations. Proof: Use the cancellation lemma as per sections 4 and 5 of Rasmussen’s thesis, but onlyfor those elements with the same bifiltration indices. ♦ We note that since the knot Floer spectral sequence for k B (0 , 0) collapses at E , theuse of the above lemma for the I -filtration means that ⊕ j C ′ ij is isomorphic to the knotFloer homology for the summands in the cube complex corresponding to that I -value. Inparticular, there are no differentials keeping I fixed, and reducing Ψ. For lack of a bettername, we will also call this reduced complex X ( L ), or just X . As a result, E I ( X ) ∼ = X forthe filtration from I . Since X is bi-filtered chain homotopy equivalent to X ( S ), it too isquasi-isomorphic to the chain complex for d CF ( − Σ( L )) by a Ψ-filtered map.We begin with a little notation: we let X j be the sub-complex of X with Ψ ≤ j . Likewise, let K j be the sub-complex of the reduced Khovanov homology with the same condition. Nowthe I -filtration – from the flattened cube– filters these sub-complexes and their quotientcomplexes. Corollary 2. Suppose there exists a n such that (1) ψ ( L ) is exact in K n (2) The I -induced spectral sequence on X n /X − g collapses at E .then c ( ξ ) = 0 . The second condition, of course, makes some complex computed from the knot Floer chaingroups isomorphic to the corresponding complex computed from the skein Khovanov chaingroups Proof: Suppose ψ ( L ) has the bifiltration value ( I ψ , − g ). If we try to compute the ho-mology of X n using the I -filtration, then ψ generates the only group in the Ψ-filtrationlevel − g . Since ψ is exact in K n , there is some element with I -filtration I ψ − K n is ψ (recall the differential increases I-values). This element, ν , may bea linear combination of elements with many different Ψ values. We note that ν is closedand not exact in E ( X n /X − g ) as a chain complex computing E . It is closed since theonly non-zero portion of ∂ Kh ν is in X − g . It is not exact since it would need to be thedifferential of something with higher I -filtration, and for those elements the differential,which is given by the Khovanov differantial, is the same as in E ( X n ); however in E ( X n ), ν is not closed and hence is not exact. Thus [ ν ] will be non-zero in E ( X n /X − g ).Consider C i to be the sub-complex of X n with I -filtration greater than or equal to i . Wehave the commutative diagram represented in Figure 6, to which the remaining argumentrefers. Here F is the homology of X − g , Q c is the quotient complex of X n by C I ψ +1 and Q is the quotient complex by { Ψ ≤ − g } ∪ { I ≥ I ψ + 1 } . The 0 in the upper left comes fromthe observation that there are no generators in X with Ψ ≤ − g and I ≥ I ψ + 1. The 0 onthe map in the upper right indicates that it will generate the trivial map in homology dueto ν . From now on we let X ′ = X n /X − g . In an earlier version of this paper, the author incorrectly asserted that the vanishing of ψ is enough toconclude that c ( ξ ) also vanishes. John Baldwin, [2], pointed out the error and has since discovered exampleswhere c ( ξ ) is non-zero despite ψ vanishing in the reduced Khovanov homology. N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 25 00 0 C I ψ +1 C I ψ +1 X n /X − g X n F QQ C F ............................................................................................... ........................................................................................................... ............ .......................................................................................................................................................................................................................................................... Id ............................................................................................................................. ................................................................................................................. ........................................................................................................... ...................................................................... ............ .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ................................................................................................................. ............ Id ................................................................................................................. ....................................................................................... ............ ......................................................................................................................................................................................................... .......................................................................................................................................................................................................................................................... ................................................................................................................. ............................................................................................................................. ............................................................................................................................. ............ Figure 6. An element of H ∗ ( Q ) in filtration level I ψ − E ( X n /X − g ). Furthermore, the argument above shows that [ ν ] = 0 in H ∗ ( Q ). This iscertainly true in Q c since ν has a non-trivial differential. However, if in Q there is anelement with differential equal to ν , the only other possibility is that in Q c this elementhas differential equal to ν plus something in X − g . But then ∂ = 0 on this element.Suppose, [ ν ] has non-zero image, [ ω ], under the map H ∗ ( Q ) → H ∗ ( C I ψ +1 ). If [ ω ] hasnon-zero image in H ∗ ( X n ), from the middle row, then it too must have a non-zero repre-sentative in E ( X n /X − g ), since C I ψ +1 has no representatives with Ψ-filtration − g . Butthen the induced differential from the long exact sequence implies that ∂ [ ν ] = [ ω ] in X ′ .Hence, the rank of H ∗ ( X ′ ) is strictly less than that at E , i.e. there is a non-trivial differ-ential beyond E .Given the assumptions, we must have [ ω ] = 0 in H ∗ ( X n ). Then it is the image of somenon-zero element, [ η ] of H ∗ ( Q c ). This element injects into H ∗ ( Q ) so that [ ν ] minus the im-age of [ η ] is the image of some non-zero element of H ∗ ( X ′ ). Furthermore, since the imageof [ ν ] under H ∗ ( Q ) → F is non-zero, then the map H ∗ ( X ′ ) → F is surjective. As a result,the map F → H ∗ ( X n ) is zero, but this implies that H ∗ ( X − g ) → H ∗ ( X ) is zero. The filtered quasi-isomorphism from X to \ CF K induces a commutative diagram H ∗ ( X − g ) ∼ = F H ∗ ( X ) HF K ( e B, − g ) ∼ = F HF ( − Σ( L )) ............................................................................................................................................................................. ............ · ...................................................................................................................... ........................................................................................................................................................................................... ∼ = ............................................................................................................................................................................... ∼ =thereby showing that the contact invariant vanishes. ♦ Note The purpose of the F coefficients is to connect with the extant versions ofKhovanov homology. In the end, the crucial observation is that Plamenevskaya’s elementuniquely defines the lowest filtered portion of the both the skein and knot Floer homolo-gies. As long as this remains true and there is an analogous reduced Khovanov homology,the same argument will work with other coefficients. In particular, just changing the signconventions will not change the conclusion, but there should be some sign convention liftedfrom the Heegaard-Floer world which will allow Z -coefficients. Note For a braid, L , we can lift a negative crossing or a positive crossing to neg-ative/positive Dehn twists along homologically non-trivial curves in the fiber of the openbook. These, in turn, fit into the long exact sequences of Heegaard-Floer and knot Floerhomology. One sign fits into the ∞ , , +1 sequence for the fiber framing, while the otherfits into the − , , ∞ sequence. Doing all the surgeries at the same time yields a spectralsequence as in the previous section, with the maps in the E page coming either from themaps ∞ → → ∞ from the respective long exact sequence and converging to theappropriate homology of the fibered knot. This is the same sequence as that constructedabove, only the basis for the framings has been alterred. Namely, if the framing from thecrossing is declared ∞ and the crossing is negative, then the 0 framing is ∞ in the fiberframing, and +1 is 0 in the fiber framing. The knot for the surgery is the same, a lift of anarc between two branched points.We now collect some results for quasi-positive braids. We note that for a quasi-positivebraid, the lifted contact structure is Stein fillable. We can use the above argument to re-prove that the induced contact element is non-vanishing, [10]. Let the braid be given by w σ i w − · · · w k σ i k w − k . We resolve only those crossings corresponding to the σ i k terms.For the 00 . . . b non-trivial circles. Any 1 resolutions makethe situation more difficult, but all the non-zero terms occur in higher filtration levels.Plamenevskaya’s element is then in the lowest level of the 00 . . . N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 27 shows that Plamenevskaya’s element survives in the spectral sequence and thus gives thenon-triviality of the contact element in the double branched cover.9. Knot Floer results for alternating branch loci We now explore the implications of the previous sections for knot Floer homology. First,we define some notation. Let L be a link in A × I admitting a connected, alternatingprojection to A . According to [12], the Heegaard-Floer homology of Σ( L , s ) is congruent to F for each of the Spin c structures on Σ( L ). For a Spin c structure s and a null-homologousknot K ⊂ Σ( L ) define τ ( K, s ) = min s ∈ Z (cid:8) s : d HF ( F s , s ) i ∗ −→ d HF (Σ( L ) , s ) is nontrivial (cid:9) where F s is the sub-complex of generators with filtration index less than or equal to s .Using the results of the previous sections and Lemma 7 we can prove Theorem 9.1. Let L be a non-split alternating link in A × I intersecting the spanning discfor B in an odd number of points. Then for each k there is an isomorphism \ HF K ( − Σ( L ) (cid:0) S × S (cid:1) , e B B (0 , , k ) ∼ = M i,j ∈ Z H i ; j, k ( L ) where, for each Spin c structure, the elements on the right side all have the same absolute Z / Z -grading. Together these isomorphisms induce a filtered quasi-isomorphism from the E -page of the knot Floer homology spectral sequence to that of the skein homology spectralsequence. Thus the knot Floer spectral sequence collapses after two steps. Furthermore, forany s ∈ Spin c (Σ( L )) we have that τ ( e B, s ) = 0 where e B is considered in Σ( L ) . The content of this theorem is that all the knots e B have the same knot Floer propertiesas alternating knots in S , and their knot Floer homology (over all Spin c structures) isdetermined by the skein homology, and the G¨oretz matrix of L , when applicable. Thelast is used to calculate the signature, through a formula of C. Gordon and R. Litherland,and the Heegaard-Floer invariants, d ( s ), for s ∈ Spin c ( − Σ( L )), [12], which determine theprecise absolute grading for the homology groups. However, it seems difficult to recoverdata about individual Spin c structures from the Khovanov formalism. Proof of theorem 9.1: We have established that there is a spectral sequence startingat the right side of the isomorphism and converging to the left side. The right side is the E page of this spectral sequence. The E page is computed using maps between resolutionsdiffering in at least two positions. Thus the maps will necessarily increase the i grading by2. However, the right side is supported in those triples satisfying k − j + 2 i = σ ( L ). If i in-creases by δ i and k stays fixed, then j must increase by 2 δ i . We compare this to the gradingin the knot Floer cubical spectral sequence which reflects the absolute grading shifts in thelong exact sequences. For example, in the Heegaard-Floer long exact sequence for surgeryon an unknot, we have Z → Z − ⊕ Z . Thinking of this as arising from a 0 → change, we would shift the right side so that the q -gradings would be preserved. In ourcase, we would shift up by . In general, q corresponds to a shift by in the absolute grad-ings. In the mapping cone construction, we shift the right side down by 1 so that the chainmap contributes to a − S × S ’s and our surgery circles as either creating or destroying one of the summands,this same calculation can be applied throughout the cubical complex to obtain a coherentrelative grading. Keeping track of the shifts yields that we can measure the relative gradingin Heegaard-Floer homology by ∆ j − ∆ i where ∆ is a change in the specified index. Forthis grading the differential at the E page is a − E page, if we fix k and consider triples with k − j + 2 i = σ ( L ) we see thatthe relative grading is 0 between generators on this plane. All higher differentials must be − k grading at E . After the E page if ∆ j − ∆ i = − k = − 2. Note that this is the shift in the additional termsdefining the spectral sequence converging to Khovanov homology.This is not the difference in the absolute gradings on − Σ( L ) as the Spin c structures havedifferent invariants, d ( s ). However, it does return the difference between e gr ( x ) − d ( s ) − s ( x ) = s . To do this measure from the element which generates d HF ( − Σ( L ) , s ) tensored with Θ ++ ∈ d HF ( (cid:0) S × S (cid:1) ). The first must exist in a single( i, j )-pair by the results of E. S. Lee, [8], and the comments in the next paragraph. Notethat the generator of d HF ( − Σ( L ) , s ) lies in the even absolute Z / Z -grading, and thus therelative grading above will be correct for Euler characteristic calculations, up to sign.The spectral sequence on the reduced skein homology collapses at its E -term (the E termin the sequence we are considering in this proof). At that stage we recover d HF ( − Σ( L )), asthe reduced Khovanov homology has total rank given by det( L ). By Lemma 7, the spectralsequence on the Khovanov skein homology is quasi-isomorphic to that on the knot Floerhomology of e B B (0 , 0) in − Σ( L ). This allows us to draw the conclusion concerning τ .Namely, the Heegaard-Floer homology of − Σ( L ) (cid:0) S × S (cid:1) will have the form e H ⊗ V ⊗ and will lie on four lines j − i = − σ ( L ) (with multiplicity 2) and j − i = − σ ( L ) ± 2. Whenwe factor out the V ⊗ , we have the reduced homology lying on k = σ ( L ) + j − i = 0. Sincethere is only one grading in each filtration level in the knot Floer homology, this impliesthat τ ( e B ) = 0 from which the result follows. ♦ We can also derive some information about e B for the branch loci depicted in Figure 1,regardless of whether L is alternating. In particular, e B ⊂ S in these cases and Lemma 9. For e B coming from the branch loci depicted in Figure 1, τ ( e B ) = − T ( L ) Proof: Add two non-trivial, non-interacting unknots to L and mark one of these. Thereis then a spectral sequence converging to the knot Floer homology of e B B (0 , 0) from H ∗ ; ∗∗ ( L ) ⊗ V . Consider an element in the subcomplex corresponding to knot filtrations N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 29 less than or equal to τ ( e B ) − −− ∈ d HF ( S × S ) under inclu-sion of the subcomplex. Then there is a element with k -gradings less than or equal to2 τ ( e B ) − L is an unknot, Θ −− is the element u − ⊗ v − . Therefore, this same element willsurvive the spectral sequence from the skein homology to the Khovanov homology. Hence T ( L ) − ≤ τ ( e B ) − − T ( L ) ≤ τ ( e B ). This is also true for B whence − T ( L ) ≤ τ ( e B ).Therefore, − T ( L ) ≥ τ ( e B ) as well. ♦ These results hold in slightly greater generality. In the sequel to this paper an argument isgiven which holds for a broader class of links, similar to the quasi-alternating links of [12].This is the smallest subset of links in A × I , denoted Q ′ , with the property that(1) The alternating, twisted unknots, linking B an odd number of times, are in Q ′ .(2) If L ⊂ A × I is a link admitting a connected projection to A , with a crossing suchthat • The two resolutions of this crossing, L and L , are in Q ′ and are connectedin A , and • det( L ) = det( L ) + det( L )then L is in Q ′ The alternating L used above are in Q ′ , and the elements of Q ′ when considered in S areall quasi-alternating as in [12]. For this class of links Wehrli’s algorithm terminates at thebase cases of our induction, from which the conclusion in the theorem can be drawn. Forbraids in Q ′ we can be more precise about Plamenevskaya’s element: Corollary 3. Let L be in Q ′ . If the element e ψ vanishes in the reduced Khovanov homology,then c ( ξ ) = 0 . Proof: This corollary follows from the non-vanishing result in Section 8 since the spectralsequence for X − g /X − g collapses according to theorem 9.1. However we need to verifythat, e ψ is zero in Kh − g . The only difficulty arises if there is a ν whose Khovanov differen-tial is e ψ and ν = P ν i where ν i is in Ψ-filtration level 2 i . Since the Khovanov differentialreduces Ψ by at most 2, this requires the i indices to range from 1 − g to l , and for there tobe a summand for each index in the range. As we collapse the complex along differentialspreserving the Ψ-filtration level, the complex stabilizes at E , and the structure describedabove yields a differential from ν l to e ψ . However, we know that at E k − j + 2 i = σ ( L ),and ν l and e ψ must have the same j value since they are linked by Khovanov differentials.In addition, the change in i is an increase of 1 from ν l to e ψ . This implies that k mustdecrease by 2, and thus l = 1 − g as required. ♦ Examples Example 0: Let L be a non-split alternating link and suppose B is a meridian of one ofthe components. Then e B is an unknot in − Σ( L ) since the spanning disc lifts to a disc. B L B L Figure 7. The diagram for example 1 is on the left; that for example 2 ison the right.Mark the link as above, then the reduced skein homology after the final shifting agreeswith the reduced Khovanov homology. On the other hand, the knot Floer homology of thisunknot is just d HF ( − Σ( L )) in filtration level 0. The equivalence of these two groups is aconsequence of [12]. In this sense, theorem 9.1 is a generalization of the result in [12]. Example 1: See Figure 7 for the diagram. Here L is an unknot in S , so e B is a knot in S as well. Untwisting and taking the branched double cover (or using symmetry betweenthe two components) shows that e B is the knot:This is the alternating knot, 6 , with signature equal to 0. The main result in [9] nowverifies the knot Floer conclusions of theorem 9.1. Furthermore, the Alexander polynomialis − T − + 5 − T . We content ourselves with a direct verification of the rank of thehighest filtration level. Only resolutions with three non-trivial circles contribute to thislevel. These resolutions and the associated generators are: v + ⊗ v + ⊗ v + v + ⊗ v + ⊗ v + v + ⊗ v + ⊗ v + ⊗ W N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 31 The maps from 010 and 001 to 011 both take v + ⊗ v + ⊗ v + to v + ⊗ v + ⊗ v + ⊗ w − , and thustheir sum is closed, as is v + ⊗ v + ⊗ v + ⊗ w + . The latter is two q -gradings above the formerclosed element, but it also has one more 1-resolution. Shifting q down by 2 decreases thehomological grading by 1 when identifying with knot Floer homology. Thus, these genera-tors are in the same grading in the knot Floer complex. This confirms theorem 9.1 for thehighest filtration level (modulo some shifting). Example 2: See Figure 7 for the diagram. Here L is the figure 8 knot, 4 , whose brancheddouble cover is L (5 , e B is a genus 1 fibered knot in L (5 , Z in filtration levels ± τ ( e B ) = 0, as this implies that there is one Spin c structure where the knot Floer homology is that of 4 . We give a non genus 1 example later.The monodromy for this knot is (cid:0) γ γ − (cid:1) where γ i is a positive Dehn twist around astandard symplectic basis element for H ( T − D ). The monodromy action on H and theAlexander polynomial associated to the Z -covering from the fibering are computed to be A = (cid:20) (cid:21) = ⇒ det( I − tA ) . = ∆ e B ( t ) = − T − + 7 − T where we have symmetrized and normalized det( I − tA ) according to the convention in thesecond appendix. We can compute the skein homology directly, but instead we use thetheory to compute it from the polynomial V ( t, q, x ). This is not quite as direct as it mayseem. The polynomial satisfies V ( t, q, x ) / ( qx + q − x − ) = t − q − + t q + 2 t − q − + 2 tq + q x + 1 + q − x − + t + 1From our conventions, we should add two non-trivial strands, and at the end factor out V ⊗ to get to the knot Floer homology. However, adding a marked non-trivial circle and anon-trivial circle amounts to multiplying qx ( qx + q − x − ) V ( t, q, x ), so will equal the aboveafter shifting and removing V ⊗ .Note that the second to last term does not satisfy k − j + 2 i = 0 and so should notappear in the homology. It cancels with one of the 1’s. In fact, this can be seen by re-solving crossings in turn, until one arrives at a the closure of σ ∈ B , where we use thehomology calculation from previously (the same cancellation occurs there) and then buildup the homology independently. When all is completed we obtain the following diagram on the ( j, k )-axes (the subscripts are the values of i ). − − − − − − F − F − F F F F − F Note that if we shift the elements to j = 0, decreasing i by 1 each time j decreases by 2,every group in the same horizontal row shifts to the same grading. Note also that the ranksafter shifting horizontally reflect the coefficients of the Alexander polynomial; and, up toa minus sign, the Z / Z -gradings are correct. Furthermore, if we consider the Ψ-filtration,in the E ∞ -page of the spectral sequence there will be five terms on the k = 0 horizontalline, correponding to the five Spin c structures on L (5 , Spin c structure and then use the Goeritz matrix for4 to complete the absolute grading calculations. To do this we should use the more re-fined torsion, ˇ τ ( Y − K ), in our Euler characteristic computations, [5]. We complete thisargument in the sequel to this paper. Comparing the two will show that the Z / Z -gradingsfrom the knot Floer homology correspond to those from the skein homology. However, thecorrespondence only occurs when we add over all Spin c structures and all q -gradings. N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 33 Appendix A. Homological Algebra All coefficients are taken in F , hence the difference from the usual signs. However, every-thing can be adapted to work with coefficients in Z .Let ( A, A ) and ( B, B ) be filtered differential modules. Let f : A → B be a filtered chainmap. Then the mapping cone M ( f ) inherits a filtration by declaring M i = A i ⊕ B i . Thatthe differential preserves this filtration follows from f being filtered. When undeclared, afiltration on a mapping cone complex will come from this construction. The definitionsimply that E ( M ) ∼ = MC( E ( f )).A filtered chain map f will be a 1-quasi-isomorphism if it induces an isomorphism betweenthe E pages of the spectral sequences for the source and the target. For the morphism ofspectral sequences induced by f , in which the induced maps intertwine the differentials oneach page, this implies that all the higher pages, E r , are quasi-isomorphic by the inducedmap, E r ( f ). This is probably weaker than f being a filtered chain isomorphism, but enoughfor spectral sequence computations.Let { ( A i , A i ) } ∞ i =0 be a set of filtered chain complexes with each filtration A i being boundedand ascending: A i : { } = A n i i ⊂ · · · ⊂ A ji ⊂ A j +1 i ⊂ · · · ⊂ A N i i ∼ = A i Let { f i : A i → A i +1 } be a set of chain maps satisfying:(1) f i is a filtered map for each i .(2) f i +1 ◦ f i is filtered chain homotopic to 0, i.e. there is a filtered map H i : A i → A i +2 such that f i +1 ◦ f i = ∂ i +2 ◦ H i + H i ◦ ∂ i .(3) f i +2 ◦ H i + H i +1 ◦ f i : A i → A i +3 is an 1-quasi-isomorphism.In this setting we have the lemma, following [12], Lemma 10. The mapping cone MC( f ) is 1-quasi-isomorphic to A . Proof: The hypotheses above guarantee that the maps in the proof of lemma 4.4 of [12]are filtered maps. We need only check the filtering condition for maps in and out of themapping cone, but with the aforementioned convention these are clearly filtered. In partic-ular the map ψ i = f i +2 ◦ H i + H i +1 ◦ f i is a 1-quasi-isomorphism by assumption, and thesame argument as in [12] implies that α is a quasi-isomorphism which is also filtered. Thisis not quite enough to conclude, but it does ensure that α i induces maps at each page inthe spectral sequence.The module Gr ( A i ) ∼ = ⊕ j ∈ Z A ji /A j − i inherits a differential which maps the j th graded com-ponent to itself, whose homology provides E . The maps f i induce chain maps betweenthese complexes for each grading level. Indeed each of the maps ψ i , H i , f i , etc., likewiseinduce such maps. Compositions such as f i +1 ◦ H i induce maps on the graded componentswhich are the same as the compositions for the maps induced from f i +1 and H i separately.Thus for each j , we have the situation in the lemma in [12] applied solely to the j th graded component. Applying the lemma in each grading guarantees that the map induced in thatgrading by α is a quasi-isomorphism, i.e. that the induced map on the E page is anisomorphism of spectral sequences. Thus, α induces an isomorphism from the E page for A to MC( E ( f )) ∼ = E (MC( f )), which is the desired result. ♦ As in [12], we can reinterpret this as a result on interated mapping cones. Let M =MC( f , f , f ) be the filtered chain complex on A ⊕ A ⊕ A , filtered by A j ⊕ A j ⊕ A j , andequipped with the differential ∂ f ∂ H f ∂ That this is a differential is a consequence of the assumptions made before the lemma.The lemma then implies that the induced spectral sequence on the iterated mapping conecollapses at the E term. This follows according to the following diagram:0 0 00 A j /A j − M j /M j − MC j ( f ) / MC j − ( f ) 00 A j A j − ⊕ A j ⊕ A j A j ⊕ A j A j − A j − ⊕ A j − ⊕ A j − A j − ⊕ A j − 00 0 0 .................................................. .................................................. .................................................................................................... .................................................. .................................................................................................... .................................................. .................................................................................................... .................................................. ...................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................. ....................................................................................................................................................................................................................... ............ .......................................................................................................................................... ................................................................................................................................................... ................................................................................................................ ............ ................................................................. ..................................................................................................................... ................................................................. ............ ............................................................................................................ ................................................................................................................................................................................................... ....................................................................................................................................................................... ............ where the top two rows are exact, and all the columns are exact. The nine lemma nowguarantees that the bottom row is exact, and each of the maps is a chain map. In the longexact sequence from the bottom row, there is one map guaranteed to be an isomorphismby the lemma. Consequently, the groups in E ( M ) are trivial. N KNOT FLOER HOMOLOGY IN DOUBLE BRANCHED COVERS 35 References [1] M. Asaeda, J. Przytycki, & A. Sikora, Categorification of the Kauffman bracket skein module of I -bundles over surfaces . Alg. & Geom. Top. 4:1177-1210 (2004).[2] J. Baldwin, Private communication .[3] D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial . Alg. & Geom. Top. 2:337-370(2002).[4] A. Champanerkar & I. Kofman, Spanning Trees and Khovanov homology . math.GT/0607510 v1[5] J. E. Grigsby, Knot Floer homology in cyclic branched covers . Alg. & Geom. Top. 6:1355–1398 (2006).[6] S. Jabuka & T. Mark, Heegaard Floer homology of certain mapping tori . Alg. & Geom. Top. 4:685-719(2004)[7] M. Khovanov, A categorification of the Jones polynomial . Duke Math. J. 101(3):359–426 (2000).[8] E. S. Lee, An endomorphism of the Khovanov invariant . Adv. Math. 197(2): 554-586 (2005).[9] P. Ozsv´ath & Z. Szab´o, Heegaard-Floer Homology and Alternating Knots . Geom. Topo. 7:225-254 (2003).[10] P. Ozsv´ath & Z. Szab´o, Heegaard Floer homology and contact structures . Duke Math. J. 129(1): 39-61(2005).[11] P. Ozsv´ath & Z. Szab´o, Holomorphic disks and knot invariants . Adv. Math., 186(1): 58-116 (2004).[12] P. Ozsv´ath & Z. Szab´o, On the Heegaard Floer homology of branched double covers . Adv. Math. 194(1):1-33 (2005).[13] O. Plamenevskaya, Transverse knots, double covers, and Heegaard Floer contact invariants .math.GT/0412183 v1[14] O. Plamenevskaya, Transverse knots and Khovanov homology . Math. Res. Lett. 13(4): 571-586 (2006).[15] A. Shumakovitch, Torsion in the Khovanov homology . math.GT/0405474 v1[16] S. Wehrli, A spanning tree model for Khovanov homology . math.GT/0409328 v2 Department of Mathematics, Michigan State University, East Lansing, MI 48824 E-mail address ::