aa r X i v : . [ m a t h . G T ] J un ON KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS
LAWRENCE ROBERTS Introduction
Consider a link in S for which one specified component, the axis, is an unknot. We denotethis link by B ∪ L where B is the axis of L and assume throughout that L intersects thespanning disc of B in an odd number 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 . The author began studying this situation in [14] where a connectionto Kohvanov homology is described. In this paper, we wish to use the same approach tostudy the special case where L is a braid in the complement of B , and derive the completeknot Floer homology for a myriad of fibered knots.We start by revisiting the main result of [14] and proving it in a purely Heegaard-Floer man-ner, in order to use Z -coefficients rather than Z / Z -coefficients as in [14]. For L alternatingfor the projection A × I → A this yields Theorem 1.
Let L be a non-split, alternating link in A × I , with det( L ) = 0 , and whichintersects the spanning disc for B in an odd number of points. Then the Z / Z -gradedknot Floer homology ⊕ i ∈ Z \ HF K (Σ( L ) , e B, F, i ) is determined by a certain Turaev torsion, ˇ τ (Σ( L ) − K ) . Furthermore, for each Spin c structure, s , on the L -space Σ( L ) we have τ ( e B, s ) = 0 . The author was supported in part by NSF grant DMS-0353717 (RTG).
First, we review the definition of ˇ τ from [4]. We will then use the above theorem to analyzethe Heegaard-Floer homology of fibered three-manifolds whose monodromies can be repre-sented as branched double covers of alternating braids. This is followed by several exampleswhich should clarify the approach and can be read after the proof of the main theorem. Inthe final section, we concentrate on deriving results in Heegaard-Floer homology similar tothose of P. Seidel, I. Smith, and especially E. Eftekhary, [3], for the Floer cohomology ofsymplectomorphisms of surfaces. In particular, we will prove Proposition 1.1.
Let M φ be the fibered three manifold determined by a fiber F g and mon-odromy D n · · · D n g g where n i ≥ and the Dehn twists occur along the linear chain of loopsdepicted in section 4. Let S be a collection of loops in F consisting of n i parallel copies ofthe loop γ i in the linear chain. Then HF + Z / Z ( M φ , s g − ) ∼ = H ∗ ( F \S ) as H ∗ ( F, Z / Z ) -modules, where the action is by cup product on the right side of the iso-morphism and by the H -action on the left. The Heegaard-Floer group is the direct sum ofthe homologies over all Spin c structures pairing with the fiber to give g − . By duality, there is a corresponding theorem when n i ≤ i .2. Background on Alexander polynomials
Let Y be a rational homology sphere; and let K ֒ → Y be a null-homologous knot withspanning surface F . Then H ( Y − K, Z ) ∼ = H ( Y, Z ) ⊕ Z . Let G = π ( Y − K ). Byduality, F defines a cohomology class in φ : G → Z . We let e X be the Z covering deter-mined by this cohomology class, and let A φ = H ( e X, e p ). The Alexander polynomial, ∆ φ , isdefined to be the greatest common divisor of the elements of the first elementary ideal of A φ .For Y an L -space, the absolute Z / Z grading in Heegaard-Floer homology assigns each d HF ( Y, s ) ∼ = F to the even grading. Moreover, this absolute grading corresponds on d CF to that given by the local intersection number at a generator, x , between the two totallyreal tori, T α and T β . It is chosen to ensure that χ ( d HF ( Y )) = (cid:12)(cid:12) H ( Y, Z ) (cid:12)(cid:12) (in fact, for anyrational homology sphere). If we choose a Heegaard decomposition of Y subordinate to K and use the presentation of G it provides, we can recover the Alexander polynomial aboveby Fox calculus relative to the map φ . Since the local intersection numbers of the totallyreal tori correspond to the signs in the determinant employed in the Fox calculus, this willalso be the Euler characteristic of the knot Floer homology \ HF K ( Y, K, F ), taken over all
Spin C structures and using the Z / Z -grading.In fact, we may choose a unique Alexander polynomial for φ by requiring that • ∆ φ (1) = (cid:12)(cid:12) H ( Y, Z ) (cid:12)(cid:12) • ∆ φ ( t − ) = ∆ φ ( t )The first statement is true of the Euler characteristic of \ HF K ( Y, K, F ) because of theEuler characteristic properties of d HF ( Y ). The second statment is true due to the identity, N KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS 3 for torsion
Spin c structures, found in [11]: \ HF K d ( Y, K, s ) ∼ = \ HF K d − m ( Y, K, J s )where J is conjugation and m = h c ( s ) , F i (assuming K = ∂F as an oriented knot). On Y the conjugation of relative Spin c structures also conjugates the underlying structures on Y ;the symmetry of the Alexander polynomial exists because we sum over all Spin c structures.In [4] a refined torsion, ˇ τ ( Y − K ) ∈ Q ( Spin c ( Y )( T )), is constructed using Turaev’s formal-ism. It has the properties that( T − τ ( Y − K ) = X s ∈ Spin c ( Y ) p s ( T ) · s for p s ( T ) = X i ∈ Z χ (cid:0) \ HF K ( Y, s ; K, i ) (cid:1) T i where the Euler characteristic is taken according to the absolute Z / Z -grading. In partic-ular, this torsion allows us to distinguish the individual Spin c structures at the expense ofa substantial increase in computational difficulty. It is related to the previous Alexanderpolynomial by X s ∈ Spin c ( Y ) p s ( T ) = ∆ φ ( T )To construct this element we need the map ǫ : π ( Y − K ) → H ( Y ; Z ) and a cell complexdecomposition of Y − K . This produces a presentation for the fundamental group to whichone applies Fox’s free differential calculus using the homomorphism φ ⊗ ǫ . In our case,we can obtain the right side of the equality above by considering the free differentials forgenerators other than the one from a meridian. This gives a square matrix and eliminatesthe pesky ( T −
1) factors in the torsion computations. For more details, consult [4].3.
Improving to Z -coefficients Our first goal is to give a proof of the following result, which is a more specific version ofthe theorem from [14]. A is a round annulus in R to fix the embedding of A × I in S Theorem 3.1.
Let L be a non-split alternating link in A × I intersecting the spanningdisc for B in an odd number of points. Then for each k the Z / Z -graded homology \ HF K Z (Σ( L ) , e B ; s , k ) has rank determined by the corresponding coefficient of ( T − τ (Σ( L ) − e B ) . The knot Floer spectral sequence collapses at the E page for any s ∈ Spin c (Σ( L )) ,and τ ( e B, s ) = 0 . Finally, if we filter CF K ∞ (Σ( L ) , e B ) using [ x , i, j ] → i + j , the inducedspectral sequence also collapses at the E page. Proof: as in [14] this is proved by induction on the number of crossings in L . In partic-ular, either of the resolutions of a crossing of L results in an alternating link with fewercrossings to which the result should apply. These resolutions correspond to two terms ina surgery exact sequence whose third term is the desired fibered knot. The homology ofthe last is isomorphic to the mapping cone of the former arising from the sequence. Theconsequences in general of this perspective are the subject of [14] following in the footsteps L. ROBERTS of [12]. We carry out the proof in a series of steps. For more detail on the approach see [14].
The base case of the induction. Consider knots of the form n − n ± n k ∓ n k − Since L is an unknot, the branched double cover is S . B ∪ L forms a link, which canbe described by flattening B along a plane and pulling L so that the loops linking B occurconsecutively in one direction. Thus we can untwist L at the expense of B , making L intoan unknot. Choose an alternating projection for this link and consider the branched doublecover. e B will be an alternating knot in S . The knot Floer homology has the followingproperties:(1) The grading of the knot Floer groups is determined by the filtration index accordingto F − τ ( K ). In particular, the Z / Z grading is just F modulo 2. Thus theAlexander polynomial determines the knot Floer homology.(2) Consequently, the knot Floer spectral sequence collapses after the E page. In fact,the i + j spectral sequence for CF K ∞ collapses after the E page.(3) τ ( K ) = − σ ( K ) where σ ( K ) is the signature of the knot We can calculate τ ( e B ) for the twisted unknots. Put in the form alternating B with L as the axis (reverse the process above, interchanging the roles of the two components). Takethe mirror if necessary, so that the outermost region for the projection of B will be coloredblack according to our coloring convention. Then 0 = σ ( B ) = O ( D ) − − n + where O ( D ) isthe number of black regions, [6]. A projection for e B can be obtained by stacking two copiesof the tangle picture for B determined by L and taking the closure. This has 2 O ( D ) − n + positive crossings. Thus σ ( e B ) = 2( O ( D ) − − n + ) = 0. So, our base cases all satisfy the required set of properties. We now follow Wehrli’s algorithm, [15], considered in the branched double cover.Recall that Wehrli’s algortihm starts by enumerating the crossings of L . One then proceedsthrough the crossings in order, looking at the two resolutions. If neither resolution discon-nects the underlying four valent graph determined by L , we resolve in both ways and thenproceed to iterate the algorithm on the two resolved diagrams. To be specific we call theresolutiond the 0 and the 1 resolution where −→ −→ N KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS 5
If either resolution disconnects the graph we move on to the next crossing. This produces atree of resolutions whose leaves are used in [15] to give a smaller complex for calculating theKhovanov homology. Each leaf is associated to a spanning tree for the Tait graph of L . Inthe double cover crossings correspond to ± S -valued Morse function. We can depictthis process downstairs by resolving L . From the knot Floer homology surgery sequenceapplied to each crossing according to the algortihm, we get a surgery spectral sequence, [14]. To complete the argument we consider a single surgery curve and consider thehomology of the ± a +1 j = ± a ∞ j ± a j A similar argument for − a − j = ± a ∞ j ± a j where the signs would be determined by which maps shift the Z / Z -gradings, and do notdepend on j . However, we assume that X j a ∗ j = | H ( Y ∗ ) | by our convention on Alexander polynomials. Since our manifolds are branched doublecovers over alternating links, by the L -space arguments of [12], we have that | H ( Y ± ) | = | H ( Y ∞ ) | + | H ( Y ) | (where we have reverted to the framing conventions of [12]). This factreflects the association between spanning trees for the Tait graphs of alternating links andthose of their resolutions. Since the signs are the same for all j , the only way both of thesecan be true is if the signs are positive. We know rk ± ,j ≤ rk ∞ ,j + rk ,j . On the other hand, by induction the terms onthe right are ( − j a ∞ j and ( − j a j , and rk ± ,j ≥ | a ± j | Using the identities above produces rk ± ,j = ( − j a ± j which in turn implies that a ± j is negative for odd j and positive for even j . In particular, all the homology in the j th level is contained in the same Z / Z grading.In the mapping cone construction, we must have that the chain map giving rise to the conehas E ( f ) trivial in order for the ranks to add. Therefore, the homology of ± Also by the L -space arguments in [12] the Spin c structures on the new manifoldpartition into two sets, one from each of the other two three manifolds. Since τ = 0 for allthe Spin c structures in these other manifolds, it is also 0 for all the Spin c structures here.Thus, the Z / Z -gradings are determined as in the rank argument above. Finally, we note that the maps involved induce maps on the knot Floer spectralsequence. If the new knot Floer spectral sequence has non-trivial higher differentials, the
L. ROBERTS induced isomorphisms (from the E terms being isomorphic) will force higher differentialsin one or the other of the resolved three manifolds. By induction this does not happen.Thus the spectral sequence will collapse at E . In fact, the chain maps also induce mapson the i + j spectral sequences from CF K ∞ (which converged in a finite number of steps).Again, by induction this rules out the possibility of new higher differentials. ♦ The argument parallels that of [14], where there is more detail, but yields an algorithmwe employ in the examples below. In fact, the above argument holds for a broader class oflinks, 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 i ) > i = 0 , L ) = det( L ) + det( L )then L is in Q ′ Then alternating L are in Q ′ , and the elements of Q ′ when considered in S are elements of Q . For this class of knots Wehrli’s algorithm terminates at the base cases of our induction,where one or other resolution will disconnect the diagram.4. Alternating braids and alternating mapping classes
We apply the preceding theory when L is a braid. If L has b strands, then the brancheddouble cover is fibered by genus ( b −
1) punctured surfaces. To specify the monodromieswe will consider, let γ , . . . , γ b − be the curves depicted as: γ γ γ γ b − γ b − Let F be a surface of genus g >
1. Let S = { S , . . . , S m } be simple closed loops in F .Assume that loops in S intersect once transversely or not at all. Define G ( S ) to be thegraph with vertices in one-to-one correspondence with the S i and edges in correspondenceto the intersection points. This graph we shall call the intersection graph of S . When G ( S )is linear and the loops are non-separating, as above, we may use the theory in the previoussections to compute the knot Floer homology. Definition 4.1.
Let δ i denote a positive Dehn twist around γ i . An element φ ∈ Γ g will becalled alternating if it can be represented as a product δ ν i δ ν i · · · δ ν k i k where (1) If i j = i l for some j and l , then sgn( ν j ) = sgn( ν l )(2) If i j = i l ± for some j and l then sgn( ν j ) = − sgn( ν l ) The element φ will be called fully alternating if there is such a representative for which { i , . . . , i k } = { , . . . , g } . N KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS 7
For an alternating representative of φ , all the Dehn twists around a given circle will beperformed in the same orientation. Any φ ∈ Γ g defines an open book decomposition of athree manifold and knot, ( Y, K ), using a variation on the mapping torus construction. Wewill mainly be concerned with the knot Floer homology of the binding for fully alternat-ing mapping classes, but we can extend the results to some non-fully alternating mappingclasses by taking connect sums of bindings and copies of B (0 , S × S ’s obtained by performing 0-surgery on two of the three components ofthe Borromean rings.Assume b >
1, so that F has negative Euler chacteristic. The set γ , . . . , γ b − is aset of essential, simple closed curves which intersect efficiently and fill the surface (theircomplement is a boundary parallel annulus). For an alternating mapping class, the sets G = { γ , γ , . . . , γ b − } and D = { γ , γ , . . . , γ b − } satisfy the criteria for the main resultof [13]. φ may be reducible, but by [13] each component map is either the identity or ispseudo-Anosov. If φ is fully alternating, then it is pseudo-Anosov. Corollary 1.
Let ( Y, K ) be a pair such that Y is an open book with binding K , abstractpage F , and fully alternating monodromy. Let A be the induced mapping on H ( F ; Z ) . If det( I − A ) = 0 , then Y is an L -space with det( I − A ) Spin C structures. Furthermore, the Z / Z -graded knot Floer homology is determined by ˇ τ ( Y − K ) . For each Spin c structure on Y , τ ( K, s ) = 0 . Proof: ( Y, K ) is the branched double cover of ( S , B ) over an alternating braid σ ν i σ ν i · · · σ ν k i k .Since the monodromy is fully alternating, the branch locus is connected. By the secondappendix the Alexander polynomial can be determined from det( I − tA ), and det( I − A ) = ± (cid:12)(cid:12) H ( Y ; Z ) (cid:12)(cid:12) . Thus Y is a rational homology sphere. We now apply the results of [12]to conclude that Y is an L -space, and then apply theorem 3.1 to compute the knot Floerhomologies. ♦ Let ∆ = a + P ni =1 a i ( T i + T − i ) be the Alexander polynomial for the fully alternatingmonodromy φ . Define the torsion coefficients for the binding, e B , by t s = ∞ X j =1 ja | s | + j then we can prove Proposition 4.2.
Let ( Y, K ) be an open book with fully alternating monodromy and pagesof negative Euler characterisitic. Suppose further that det( I − A ) = 0 . Let Y K be thefibered three manifold obtained by page framed surgery on K . Then, for all s > we havea Z [ U ] -module isomorphism M { s : h c ( s ) , [ b F ] i =2 s } HF + ( Y K , s ) ∼ = Z b s L. ROBERTS where the the Z / Z grading of the right hand side is ( s mod 2) , [ b F ] is the class in H ( Y K ; Z ) for a capped page, and b s = ( − s +1 t s ( K ) Proof:
Since the knot Floer homology of the binding behaves like an alternating knot in S we mimic the proof of theorem 1.4 in [9]. There are a few changes due to the lack ofabsolute grading information. First, note that for p >>
0, the isomorphism CF + ( Y p ( K ) , [ s ]) −→ C { max( i, j − s ) ≥ } from [11] still applies in our setting, since we have a prescribed spanning surface. If wefilter the left side by [ x , i ] → i and the right side by [ y , i, j ] → i + j , then both sidesare filtered complexes. The chain isomorphism above takes [ x , i ] to a sum of terms such as[ y , i − n w ( ψ ) , i − n z ( ψ )] with n w ( ψ ) − n z ( ψ ) = s . Now i − n w ( ψ )+ i − n z ( ψ ) = 2 i + s − n w ( ψ ) ≤ i + s . Thus the chain isomorphism is filtered and we have a spectral sequence morphismwhich converges to an isomorphism. For each i + j value, the E term on the right is a sumof (shifted) knot Floer homology groups for the binding. From 3.1 these have the propertythat gr Z / Z ( x ) = F ( x ) mod 2 gr Z / Z ([ x , i, j ]) = i + j mod 2The homology groups are constructed through the long exact sequence. At each stage ofthe long exact sequence there are chain maps which we apply to CF K ∞ to obtain spectralsequence morphisms for the i + j filtration. If in CF K ∞ there is a differential which isnot ( − ,
0) or (0 , −
1) then it must not induce a higher differential past the E term. Thisoccurs because the E term is isomorphic to a lower group in the resolution tree, and byinduction these do not have higher differentials after the E term (they collapse at E evenin HF K ∞ ). Thus, we need only consider up to the E terms to calculate the homologiesof these complexes up to isomorphism.We can now proceed as in [9], replacing C { max( i, j − s ) ≥ } , and every such complex,with E { max( i, j − s ) ≥ } . Furthermore, we split these complexes up according to the Spin c structure on Y . We further decompose each of these E pages into subgroups E k for k ∈ Z by using those generators with absolute grading given by gr Q ([ x , i, j ]) = i + j + d ( s ) + k These are subcomplexes of E ( s ) since only the ( − ,
0) and (0 , −
1) differentials will con-tribute. For each E k we apply the argument from [9], noting that only for k = 0 will weobtain a tower T + . Every other k merely produces some finite group in a specific grad-ing ( s − k + d ( s )). The grading subscripts in the reduced homologies in [9] will nowonly record the Z / Z -grading. Now, however, we obtain for large positive surgeries on thebinding, E s { max( i, j − s ) ≥ } ∼ = T + d ( s ) ⊕ Z m ( s ,s ) s − for each of the Spin c structures where d ( s ) defines the 0 in the Z / Z -grading. So thereduced homology in each of these homologies occurs in grading s − N KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS 9
B L B L Figure 1.
The diagram for example 1 is on the left; that for example 2 ison the right.the spectral sequence for
CF K ∞ collapse at this point, the tower in E is the tower in E ∞ . The reduced homology all has the same Z / Z -grading, and thus the spectral sequencecollapses entirely. Using the long exact sequence, · · · −→ HF + ( Y, s ) F −→ HF + ( Y K , s s ) F −→ HF + ( Y p ( K ) , s s ) F −→ · · · we know that HF + ( Y, s ) ∼ = T + d ( s ) is in the even absolute gradings, as is the tower in HF + ( Y p ( K ) , s s ) ∼ = E s { max( i, j − s ) ≥ } with an absolute grading shift. The map F is modelled on the surjection C { max( i, j − s ) } → C { i ≥ } . Since τ ( e B ) = 0 and s > HF + ( Y, s ). F corresponds to a positivedefinite cobordism and therefore reverses the Z / Z -gradings, implying that F (a cobordismwith a new 0 framed homology class) and F preserve them. Therefore, the Z m ( s ,s ) s − termgives rise to a Z m ( s ,s ) in Z / Z -grading given by s modulo 2.Adding these over all Spin c structures provides the identification with the torsion coef-ficients, by way of the result that − t ( Y K , s ) = χ ( HF + ( Y K , s )) since b ( Y K ) = 1. Notethat we use the fact that the projection of τ ( M ) under projection to Z [ H / Tors] is still∆ K / ( t − . In fact, the values of m ( s ,s ) should be given by the Turaev torsion directly. ♦ Examples
Example 1:
See Figure 1 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 3.1. We note for later that the Alexander polynomial is − T − + 5 − T . Example 2:
See Figure 1 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 infiltration levels ±
1. The real content of the theorem here is that τ ( e B ) = 0, as this impliesthat there is one Spin c structure where the knot Floer homology is that of 4 . We give anon 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 our convention. Infact we should use the more refined torsion, ˇ τ ( Y − K ), in our Euler characteristic compu-tations, [4]. This we now proceed to calculate.The fundamental group of Σ( L ) − e B can be computed using the basis for π ( F ) above.The action of ( D γ D − γ ) on the two elements generating this free group is γ −→ γ γ γ γ γ γ γ = R ( γ ) γ −→ γ γ γ γ = R ( γ )These provide the relations γ − tR ( γ ) t − and γ − tR ( γ ) t − for the fundamental group.For the choice above, we obtain the map on homology e → e + 3 e and e → e + 2 e .The quotient Z / ( I − A ) has a basis given by (5 ,
0) and (3 , H (Σ( L )) isthereby given as γ → e and γ → e − (switching to exponents) with e = 0. We now applyFox calculus to the relations, and then map to H (Σ( L ) − e B ) using the previous map and t → T . We illustrate with one calculation:( ρ ⊗ ǫ )( ∂ γ R ) = ( ρ ⊗ ǫ )( γ − t (cid:0) γ + γ γ γ + γ γ γ γ γ (cid:1) )= e − t ( e + e · e − · e + e − ) = t (1 + e + e )The overall matrix (removing the column for the derivatives related to t ) is (cid:20) − e + T (2 e + e + e + e ) T (1 + e + e ) T ( e + e + e ) − e + T ( e + e ) (cid:21) which has determinant − T − T e + (1 − T + T ) e − T e − T e . Symmetrizing in T produces the correct Euler characteristic up to signs. This determines the ranks in each ofthe filtration indices for each of the Spin c structures. N KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS 11 +1+1 − − − − − ∞ +2 − − − Figure 2
This fibered knot can also be seen by surgery on the diagram for B (0 ,
0) as in Figure 2.The box around +3 indicates that we will use it for the surgery sequence with framings ∞ , +2 , +3. For ∞ we obtain a knot in L (2 ,
1) identical with B ( − , L ( − , L (5 ,
2) has five
Spin c structures, the sequence forthe \ HF K splits (all the homology from L ( − ,
1) must map to that for the fibered knot in L (5 , B ( − ,
1) has homology Z in filtration level 0 for one Spin c structure,and homology identical with \ HF K (4 ) in the other. This follows from a Borromean ringscalculation or can be found in [5]. Thus all the terms in filtrations other than 0 shouldoccur for a single Spin c structure (the one fixed under conjugation). Example 3:
Consider the situation in section 5.2 of [5]: a genus 1 fibered knot withmonodromy D nγ D mγ with m · n <
0. This is the branched double cover of the closure ofthe three stranded braid σ n σ m . Assume for now that m <
0. Then the action of themonodromy on H ( F ) is given by A = (cid:20) n (cid:21) (cid:20) − m (cid:21) = (cid:20) − mn − mn (cid:21) = ⇒ det( I − tA ) . = ∆ e B ( t ) = − T − + (2 − mn ) − T
12 L. ROBERTS
By theorem 3.1, the Alexander polynomial determines the knot Floer homology groups.From Heegaard Floer homology, there must be an F in filtration level 0 for each of the (cid:12)(cid:12) m · n (cid:12)(cid:12) Spin c structures on Σ( L ) = L ( m, L ( n,
1) (the closure of the braid is a connectsum of a (2 , m ) torus link and a (2 , n ) torus link). This implies that the knot Floer homologyis given by \ HF K (Σ( L ) , e B, j ) ∼ = F j = 1 F (2 − mn ) j = 0 F j = − j = 0 level in the even grading. With the observation about absolute gradings inthe proof of theorem 3.1, we recover Lemma 5.5 of [5] up to the decomposition into Spin c structures.Once again, we can compute ˇ τ (Σ( L ) − e B ). The map on π ( F ) is γ −→ (cid:0) γ n γ (cid:1) | m | γ = R ( γ ) γ −→ γ n γ = R ( γ )Following the procedure above produces( T − τ (Σ( L ) − e B ) = − T − + h (1 + e . . . + e n − )(1 + e + . . . + e | m |− ) + 2 i − T where e n = e | m | = 1 and the map to H (Σ( L ) is given by γ → e and γ → e . Thus,there is one Spin c structure where the knot Floer homology is that of 4 and the rest aretrivial. This is most of the result in [5]. Example 4:
We outline an example for a pseudo-Anosov mapping class on a genus-twosurface with boundary. Consider the five stranded braid σ − σ − σ σ σ − . This correspondsto the monodromy D − γ D − γ D γ D γ D − γ . L then consists of a chain of four unknots withlinking numbers −
1, +1, and − π ( F ) is γ −→ γ γ γ −→ γ − γ − γ γ γ γ − γ − γ γ γ γ −→ ( γ − γ − γ γ γ ) ( γ γ γ − γ − γ − ) γ γ ( γ γ γ − γ − γ − ) γ γ γ −→ γ γ γ γ − γ − γ − γ γ γ − γ − γ − γ γ which yields the following map on H ( F ): A = − −
22 5 − − − − = ⇒ ∆ e B ( T ) . = T − T + 34 − T − + T −
2N KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS 13 U U
Figure 3.
The resolution tree for σ − σ − σ σ σ − from Wehrli’s algorithm.One should think of these as standing in for their closures. We depict theresults after resolving and simplifying. At each leaf we obtain a twistedunknot, and the label is the knot, e B , found in the branched double coverover this unknot (up to mirrors). Note that there is one 8 label, two 6 labels, three 4 labels, and two unknots. This should be compared with theresult in Example 4. To distinguish the
Spin c structures, we note that det( L ) = 8 and Z / ( A − I ) ∼ = Z / e ⊕ Z / e ⊕ Z / e with e = e (where e i = [ γ i ]). Indeed, Σ( L ) = L (2 , L ( − , T − τ (Σ( L ) − e B ) requires a great deal more effort, but ultimately yields: (cid:0) T − T + 13 − T − + 1 (cid:1) + (cid:0) − T + 5 − T − (cid:1)(cid:0) e + e (cid:1) + (cid:0) − T + 3 − T − (cid:1)(cid:0) e + e e + e e (cid:1) + (cid:0) e e + e e e (cid:1) We make three observations: 1) setting e i to 1 returns ∆ e B ( T ), 2) there are non-trivialphenomena in the knot Floer homology of more than one Spin c structure, and 3) the coef-ficient for each Spin c structure is the Alexander polynomial of a τ = 0 alternating knot in S (the first is that for 8 , the second for 6 , and the third for 4 ). If one follows Wehrli’salgorithm, we obtain the tree of resolutions in Figure 3, which demonstrates how the knotFloer homology is built out of simpler pieces.6. On a theorem of E. Eftekhary
In [3] E. Eftekhary proves the following theorem for the Floer cohomology of symplecto-morphisms:
Theorem 2. [3]
Let S be a set of simple, closed, non-separating loops in a surface, b F , eachpair of which are either disjoint or intersect transversely in a single point and such that G ( S ) is a forest. Let φ be the composition of a single positive Dehn twist along each loopin S , taken in any order. Then HF ∗ symp ( φ ) = H ∗ ( b F , S ) as H ∗ ( b F , Z / Z ) modules where H ∗ ( b F ) acts on the right side by the cup product and the leftside by the quantum cup product. There is also a version for negative Dehn twists replacing H ∗ ( b F , S ) by HF ∗ ( b F \S ) and aversion for compositions of negative Dehn twists and positive Dehn twists as long as theyoccur on separated forests. Note that we have an element of S for each Dehn twist; powersof the same Dehn twist should be construed as occuring along parallel copies, all of whichare in S , of a single curve.Due to the presumptive equivalence between various Floer homology theories, it has beensuggested that a similar property should hold for the Heegaard-Floer homology of thefibered three manifold in the above theorem. This approach questions whether the sym-plectic cohomology can be replaced by HF + ( M φ , s g − ), or some equivalent (using duality).Since HF + ( M φ , s g − ) ∼ = Z for every fibered three manifold, this is the next simplest caseto try to compute. In [5], the same statement using HF + is verified for certain genus 1fibered three manifolds. Our purpose now is to extend their results to a certain case where G ( S ) is a collection of linear chains preserved by a hyperelliptic involution. We do this byfirst computing the knot Floer homology of the binding of an associated open book. N KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS 15 A A A g − A g A A g − A g A Figure 4.
Braids whose closures are connect sums. Each small loop in theright diagram is used to connect sum to the piece to its right. The result of allthe connect sums is the picture on the left. If we assume that A , . . . , A g are connected alternating braids, each using an even number of strands,then the knot Floer homology of e B for the left diagram is determined bythe spanning trees of its Tait graph.Certain braid closures are connect sums of simpler pieces. These are depicted in Fig-ure 4, where each A i is a braid and the pieces are joined together in a staircase pattern. Inorder to have an odd number of strands intersecting the spanning disc for B , we ask thateach A i intertwines an even number of strands. Furthermore, if these pieces are alternatingthen the the number of spanning trees for each piece equals the number of Spin c structureson the double branched cover. For the connect sums in Figure 4, the number of trees isthe product of the number of trees for each simpler piece, and the same is true for Spin c structures by the standard results for gluing along spheres. A similar conclusion to theorem3.1 then holds: the knot Floer homology of the branched double cover of the axis is a directsum (over Spin c structures) of the knot Floer homologies of the double branched coversover the twisted unknots arising in the resolution tree. Furthermore, we can determinethe exact knot Floer homology by finding all the spanning tree unknots and examiningthe branched double covers (knots in S ) in those cases. This gives an effective algorithmfor determing the knot Floer homology. However, now the value of τ ( e B ) may be differentfor the different pieces, since the base cases are no longer necessarily alternating. As oneexample consider the braids σ n · · · σ n g g , where the exponents may be either positive ornegative. When all the n i ≥ g = 1 was addressed in [5], andthe technique in this paper recovers their results. For g ≥
1, tracing through the algorithm,using 0 smoothings whenever possible, shows that there is one
Spin c structure whose knotFloer homology will be identical with that of T , g +1 . The others are more involved. For example, for σ σ σ σ , the Alexander polynomial is calculated from the monodromyaction on H ( F ) to be ∆ e B ( T ) = T + 8 T − T − + T − but the knot Floer homology, summed over all the Spin c structures of L (2 ,
1) is \ HF K ( e B, j ) = Z (0) j = 2 Z ( − ⊕ Z j = 1 Z ( − ⊕ Z − ⊕ Z j = 0 Z ( − ⊕ Z − j = − Z ( − j = − Z -relative gradings for the absolute grading in each Spin c structure.This computation follows by noting that the resolution tree gives 8 unknots, 5 copies of T , , 2 copies of 5 , and 1 copy of T , , as the branched double covers of the 16 twistedunknots at the leaves.Now consider braids on 2 g + 1 strands of the form Q j w j where w j = σ i j σ i j +1 . . . σ i j +2 k j − and i j +1 > i j + 2 k j . The last condition ensures that the braid words, w j , include disjointsets of generators. The monodromy of the open book in the branched double cover isthen a series of negative Dehn twists along curves whose intersection graph forms a for-est of trees with no limbs. Furthermore at most one twist occurs along each circle. Ifwe perform fibered framed 0-surgery on the binding, the resulting fibered three manifoldsatisfies the conditions of the theorem in [3] for the Floer cohomology of symplectomor-phisms. To compute the knot Floer homology of e B in this case, we will identify how theknot in the branched double cover reflects the connect sum decomposition as just described.First, for each σ j not included in the product of the w j ’s there is a core circle, c j , inthe annulus, A , which does not intersect the diagram for L , and which is the intersection ofthe plane with a sphere in S . This sphere can be chosen to intersect B in two points. If wecut along all such spheres we obtain two B pieces and a bunch of S × I pieces. Each piececontains some portion of L and either an arc (for the B ’s) or two arcs (for the S × I ’s)of B . These pieces can be ordered from left to right in accordance with the ordering onthe braid. We now fill every S with a copy of B and complete each set of arcs from B with unknotted arcs in the new B components to obtain a knot. This realizes the pair( B, L ) as a pair connect sum along the axes of ( S , L j , B j ) where j indexes the pieces asin the braid. Furthermore, each L j has an odd number of strands in it, due to the form ofthe braid word (we can alter this to included even number of strands and reach the sameconclusion, but this requires more work). The branched double cover of ( S , L j , B j ) over L j is a copy of ( S , T , k j +1 ) where a single strand of L gives rise to an unknot.We recover the branched double cover from the pieces in the following way. On B i thereare one or two arcs which were glued in during the decomposition process. One arc for thepieces from either end of the braid; two for any piece from the interior. These lift to two or N KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS 17 four arcs in the double cover (in the case of four, the lifts alternate around the knot e B i ). Toconstruct the branched double cover, remove small ball neighborhoods around these andglue to the corresponding lift in the branched double cover of the pieces from the left andright in the braid diagram. One (or two) of these arcs glue just as connect sums of pairs(double cover, knot)to the piece on the left and/or right. To glue the other two arcs, add afour dimensional one handle and connect sum across the one handle. The boundary is theknot in the branched double cover. Note that this means that the knot sits in a connectsum of S × S ’s.We have made our choices so that there will be an even number of one handles added,say 2 m . From [11], in this setting we can peel off m copies of B (0 , ⊂ S × S as con-nect summands, leaving a connect sum of the underlying knots T , k j +1 in S . This allowsus to calculate the knot Floer homology of e B : \ HF K ( e B, g ) ∼ = Z m while the computation of \ HF K ( e B, g −
1) is slightly more complicated. We have that 2 m + P k j = 2 g . We use thegenus -1 level of each B (0 ,
0) or T , k j +1 in turn, tensored with the genus levels of all theothers. Using the lower level in a B (0 ,
0) summand produces Z ⊗ Z ⊗ · · · ⊗ Z ⊗ Z ⊗ · · · ⊗ Z = Z m − There are m of these, for a total contribution Z mm − . Using the lower filtration from a T , k j +1 summand produces Z ⊗ · · · Z ⊗ Z − ⊗ Z ⊗ · · · ⊗ Z contributing Z m − a total of s times, where there are s torus knots. Thus \ HF K ( e B, g −
1) = Z m + sm − .We can now compute the Heegaard-Floer homology of 0-surgery on e B . We use a theorem ofP. Ozsv´ath and Z. Szab´o , an account of which is given in [5]. Key to this approach is thatthe knot is in a three manifold without reduced homology (and only one Spin c structurewith any non-trivial homology). This theorem states that HF + ( Y e B (0) , s g − ) is then isomor-phic to the portion of the Z ⊕ Z -graded complex CF K ∞ described by C { i < , j ≥ g − } .For us this complex is isomorphic to Z m − Z m − ⊕ Z mm − Z m − ... Z m − ....................................................... ....................................................... ................................................... s where the arrows to the left are all isomorphisms. These arrows come from the complexesfor the torus knots, specifically from the surjective differential onto the − g ( T , k j +1 ) in-dexed summands. Thus HF + ( Y K , s g − ) ∼ = Z m − ⊕ Z m + s − m − where the gradings should nowbe taken as relative gradings. Let b F be the genus g fiber for Y K . We have that b F \ C is agenus g − P k j = m surface with s boundary components. Thus, H ∗ ( b F \ C ) ∼ = Z ⊕ Z m + s − where the first summand is H ( b F \ C ) and the second is H ( b F \ C ). Finally, the actionof H ( Y K ) / Tors, excluding the Z introduced during the surgery, is identical to that from ∼ ∼ Figure 5.
If a one resolution is employed in either of the extreme twistregions we can isotope the closure of the fully resolved diagrams so thatthere are two fewer strands. m B (0 , Z / Z ) to the standard cup product on b F . This verifiesa Heegaard-Floer analog of E. Eftekhary’s theorem for this set of braids.The theorem in [3] is slightly more general, even in our setting, allowing words of the form σ n j i j σ i j +1 . . . σ i j +2 k j − σ m j i j +2 k j − with n j , m j >
0. It is a consequence of one of the lemmas in[3], or by inspection, that any reordering of this product can be brought into the form aboveby isotopies of the braid closure. Simply start at the left hand side, just above the firstcrossing in the second column and isotope all the crossings in the first column around theclosed braid until they all lie under the crossing in the second column. Now take everythingin the first and second column and isotope around until they all lie below the crossing inthe third column. If we keep doing this we obtain a braid word of the kind at the beginningof this paragraph. In fact, this is all that can happen for a linear chain in his theorem .If the exponents are bigger than one in the interior of a word then G ( S ) is no longer a forest.Using our techniques we can generalize further, allowing words where all the exponentscan be arbitrary positive numbers: Y j σ n ij i j σ n ij +1 i j +1 . . . σ n ij +2 kj − i j +2 k j − σ n ij +2 kj − i j +2 k j − We now proceed to analyze this case.The closure of each word is subject to the analysis of the ladder braids mentioned atthe beginning of this section. We need only determine the contributions of each
Spin c structure on the branched double cover to \ HF K ( e B, k j ) and \ HF K ( e B, k j − T , k j +1 in thebranched double cover. This contributes a Z to \ HF K ( e B, k j ) and a Z − to \ HF K ( e B, k j − T , k j − if the resolution occurs in either of the extreme twist regions, see Figure 5.There are n i j − n i j +2 k j − − N KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS 19 ∼∼ Figure 6.
If the single one resolution occurs in the middle of the braid, theclosure is isotopic to the closure of a tangle with a clasp. The final diagramshows the pieces used in the plumbing decomposition of e B in the brancheddouble cover. The central piece alone will yield a copy of 5 for e B in thedouble branched cover, and this has a Z in its top knot Floer homologyfiltration level. Note that this is a positive braid, so we know that it is 5 and not its mirror.occurs in one of the interior twist regions we obtain a diagram as in Figure 6 which con-tributes Z to \ HF K ( e B, k j − n i j + l − i j + l column.To see the contibution, frist intercahnge the roles of e B and L and note that we can group aneven number of crossings from each end of the braid so as to leave precisely three crossings,two for the clasp and one more to one or other side of the clasp. In the double cover, thisgrouping corresponds to viewing e B as the plumbing of up to three objects, two torus knots T , k +1 and 5 . In addition, the plumbing occurs along genus minimizing spanning surfacesfor each of the knots. By the Y. Ni’s theorem, [8], concerning Murasugi sums of knots, thisgives the contribution as Z , where e B being a positive knot implies τ ( e B ) = g ( e B ), [7], andthus confirms the grading.When we allow two 1 resolutions not occuring in consecutive columns, or three or more 1resolutions, we can see that there is no contribution to the k j − Figure 7.
The diagram on the left depicts the situation for two non-consecutive 1 resolutions. The dotted lines trace the discs which lift to acompression disc for the fiber in the branched cover. The crossing assump-tions ensure that the discs are above the diagram. The arcs in the spanningdisc for B won’t intersect since there is at least one strand between the tworesolutions. The right diagram depicts the situation when the resolutionsare consecutive. Now the compression discs will intersect in a point. Fol-lowing the arrows will cancel the critical points in the diagram and leave acopy of σ . . . σ k .leftmost, and the strands opening up from the rightmost both will intersect the spanningdisc for B in cancelling pairs of points. Moreover, since these are not consecutive, thestructure of the braid ensures that they will be disjoint. By following a small arc betweenthese strands starting at the critical point, we obtain two arcs, necessarily disjoint, andtwo discs swept out by these arcs. These discs do not intersect L in their interiors and liftto compression discs for F in the double cover. Hence the branched double cover for theresolved diagram has minimal genus for its binding smaller than k j − T , k j − which contributes a Z to the k j − n i j + l − n i j + l +1 −
1) such contirbutions from the i j + l and i j + l + 1columns. Thus, adding up all these contributions yields that \ HF K ( e B, k j −
1) is congruentto Z − ⊕ Z T j where T j = P l ( n i j + l n i j + l +1 −
1) since the latter is equal to( n i −
1) + ( n i − n i +1 −
1) + 2( n i +1 −
1) + ( n i +1 − n i +2 −
1) + . . . + ( n i +2 k j − − \ HF K ( e B, g ) = Z m but \ HF K ( e B, g − ∼ = Z mm − ⊕ Z sm − ⊕ Z P j T j m N KNOT FLOER HOMOLOGY FOR SOME FIBERED KNOTS 21
As before, the differential in the relevant subset of
CF K ∞ carries each generator of Z s to Z , coming from the spectral sequence for T , k j +1 . So, we obtain HF + ( Y K , s g − ) ∼ = Z P T j +1 m ⊕ Z m + s − m − with action of H entirely contained in the Z m ⊕ Z mm − portion of the complex (arisingfrom the B (0 ,
0) summands). b F \ C consists of the surface in the simpler case along withnumerous squares. It is straightforward to verify that there are P T j such “squares” whicheach have rank one H -group. Again we also obtain the correspondence of the action of H with the cohomology ring.Unfortunately, not all positive braids possess the property in Eftekhary’s theorem. Thebraid σ σ σ σ σ σ σ describes an open book upon which 0 surgery has a rank 2 HF + -group for the relevant Spin c structure. This follows from the following calculation for closedbraids C ( σ σ σ σ σ σ σ ) = C ( σ σ σ σ σ σ σ ) = C ( σ σ σ σ σ σ σ ) = C ( σ σ σ σ σ σ σ ) = C ( σ σ σ σ σ σ σ ) = C ( σ σ σ σ σ σ σ ) = C ( σ σ σ σ σ σ )The results above apply to the last braid closure, and yield that HF + ( Y, s ) ∼ = 2. However, H ∗ ( b F \ C ) ∼ = Z for the first braid. The singular homology does not transform appropriatelyunder the braid equivalences for the result to hold. From the proof above, we can identifythe difficulty: the single allowable 1 resolution for the original braid, although it does notoccur at the ends of the braid, still produces a torus knot. The additional σ allows thetwo critical points introduced during the resolution to cancel with each other instead offorming a clasp. This suggests first assuming some kind of normal form for our braidsbefore proceeding further. References [1] M. Asaeda, J. Przytycki, & A. Sikora,
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Department of Mathematics, Michigan State University, East Lansing, MI 48824
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