Knot Floer homology of satellite knots with (1,1)-patterns
aa r X i v : . [ m a t h . G T ] D ec KNOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , -PATTERNS WENZHAO CHEN
Abstract.
For pattern knots admitting genus-one bordered Heegaard dia-grams, we show the knot Floer chain complexes of the corresponding satelliteknots can be computed using immersed curves. This, in particular, gives aconvenient way to compute the τ -invariant. For patterns P obtained fromtwo-bridge links b ( p, q ), we derive a formula for the τ -invariant of P ( T , ) and P ( − T , ) in terms of ( p, q ), and use this formula to study whether such pat-terns induce homomorphisms on the concordance group, providing a glimpseat a conjecture due to Hedden. Introduction
In 2001, Ozsv´ath and Szab´o defined a package of invariants for closed oriented 3-manifolds known as the
Heegaard Floer homology [15]. Later on, a bordered theoryfor the hat-version Heegaard Floer homology is developed by Lipshitz-Ozsv´ath-Thurston [10]. This theory associates certain differential modules and A-infinitymodules to 3-manifolds with parametrized boundaries, called type D ( \ CF D ) andtype A structures ( [ CF A ) respectively. For a closed 3-manifold constructed fromgluing two 3-manifolds with boundaries, the corresponding hat-version HeegaardFloer homology can be obtained by an appropriate tensor product of the type Dstructure of one piece and the type A structure of the other. Recently, Rasmussen-Hanselman-Watson gave a geometric interpretation of the bordered theory for 3-manifolds with torus boundary [2]: the bordered invariants are interpreted as im-mersed curves decorated with local systems on ∂M \{ w } , where w is a base point,and the pairing of type D and type A structures translates to taking Lagrangianintersection Floer homology of the curve-sets on the torus.The knot Floer homology is a variant of the Heegaard Floer homology definedfor null-homologous knots in closed oriented 3-manifolds, introduced by Ozsv´ath-Szab´o and independently by Rasmussen [14, 17]. One can regard this invariant asthe filtered chain homotopy type of certain Z -filtered chain complex \ CF K ( K ) asso-ciated to a knot K . The bordered theory carries through this setting: it associatesfiltered bordered invariants to knots in 3-manifolds with parametrized boundaries,and the knot Floer homology for knots in 3-manifolds constructed by glueing canbe obtained by tensoring the corresponding bordered invariants. This machinery iswell suited for studying the knot Floer homology of satellite knots . Recall given apattern knot P ⊂ S × D and a companion knot K ⊂ S , the satellite knot P ( K )is constructed by gluing ( S × D , P ) to the knot complement X K = S − nb ( K )of K so that the meridians are identified, and the longitude of S × D is identifiedwith the Seifert longitude of K . For example, the knot Floer homology of satellite knots obtained by cabling or applying the Mazur pattern were studied using thistool [8, 9, 16].As mentioned above, pairing unfiltered bordered invariants of 3-manifolds withtorus boundary may be taken as Lagrangian intersection Floer homology of curveson the torus. In this paper, we seek to explore a counterpart for the pairing of thefiltered type A structure [ CF A ( S × D , P ) and the (unfiltered) type D structure \ CF D ( X K ), and hence obtain the knot Floer homology of the satellite knot P ( K ).1.1. The main theorem.
Our main theorem will restrict to a class of patternknots called (1 , -pattern knots . The aforementioned cabling and Mazur patternbelong to this class, as well as the Whitehead double operator. Definition 1.1.
A pattern knot P ⊂ S × D is called a (1 , , α a , β, w, z ) be a genus-oneHeegaard diagram for P ⊂ S × D . Note we may view the objects ( β, w, z ) asembedded in ∂ ( S × D ), and the α a arcs correspond to the meridian µ and longitude λ of ∂ ( S × D ). So the data contained in such a bordered Heegaard diagram can beequivalently be understood as a 5-tuple ( β, µ, λ, w, z ) ⊂ ∂ ( S × D ) (Figure 11). Wewarn the reader that this set of data depends on the choice of Heegaard diagrams,and hence is not an invariant of the pattern knot. For a 3-manifold M with torusboundary, we denote the immersed-curve invariant as ( d HF ( M ) , w ) ⊂ ∂M and callit the Heegaard Floer homology of M . The main theorem is stated below. Thereaders who prefer a more visual presentation may first read Example 1.4 and thencome back to the following formal statement. Theorem 1.2.
Given a (1 , -pattern knot P ⊂ S × D and a companion knot K in S . Let ( d HF ( X K ) , w ′ ) ⊂ ∂X K be the Heegaard Floer homology of knot complement X K of K , and let ( β, µ, λ, w, z ) ⊂ ∂ ( S × D ) be the 5-tuple corresponding to somegenus-one Heegaard diagram for P . Let h : ∂X K → ∂ ( S × D ) be an orientationpreserving homeomorphism such that(1) h identifies the meridian and Seifert longitude of K with µ and λ respec-tively;(2) h ( w ′ ) = w ;(3) there is a regular neighborhood U ⊂ ∂ ( S × D ) of w such that z ∈ U , U ∩ ( λ ∪ µ ) = ∅ , and U ∩ h ( d HF ( X K )) = ∅ .Let α = h ( d HF ( X K )) . Then there is a chain complex isomorphism \ CF K ( α, β, w, z ) ∼ = \ CF K ( S , P ( K )) . Moreover, if α is connected, this isomorphism preserves the Maslov grading andAlexander filtration.Remark . When α ( K ) is not connected, the full grading information can still berecovered when provided extra data called “phantom arrows”. Roughly, these arearrows that connects different components of α ( K ) and does not alter the chaincomplex obtained by pairing. In general α ( K ) is not connected, yet it always hasa distinguished component which is gives a nontrivial homology class in H ( ∂X K ), NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , and when one is interested in computing the τ -invariant of the satellite knot, itsuffices to restrict to the distinguished component [3]. Example 1.4.
In practice, Theorem 1.2 amounts to laying the bordered Heegaarddiagram of the pattern knot over the imersed-curve diagram of the knot comple-ment. We consider the Mazur pattern M actting on the right-handed trefoil T , .In Figure 1 (a), a 5-tuple corresponding the Mazur pattern is given on the left, theimmersed curve for the trefoil complement is drawn on the right, and the pairingis given in the middle. By lifting the curves to the universal cover of the torus anddoing isotopy, the pairing diagram can be presented as in Figure 1 (b). This is theminimal intersection diagram and hence the intersection points are in one-to-onecorrespondence with elements in \ HF K ( M ( T , )). Figure 1.
The pairing diagram for \ CF K ( M ( T , ))In addition to a quick computation of the rank of the knot Floer homology group,Theorem 1.2 also gives a handy way to compute the τ -invariant of such satelliteknots. In fact, one may repeatedly isotope β ( P ) across the base point z in thepairing diagram to eliminate intersection points with minimal Alexander filtrationdifference, and in the end only one intersection point is left, whose Alexandergrading is exactly the τ -invariant of the satellite knot (Figure 10). This process isdescribed in detail in Section 4.We point out the idea for proving Theorem 1.2 using Figure 2. We work withthe universal cover C of the torus. Given any embedded Whitney disk, we canpush and collapse it to get a disk in a covering space of the bordered Heegaard WENZHAO CHEN diagram (Σ , α a , β, w, z ). This latter embedded disk gives rise to a type A operationin [ CF A ( P, S × D ), whose input of the elements in the torus algebra matches thethose coming from the arcs on the α curve, i.e. type D operations in \ CF D ( X K ).This shows the correspondence of the differentials in the Lagrangian intersectionFloer homology and in the box-tensor product. The detailed proof for Theorem 1.2is in Section 3. Figure 2.
A disk in the Lagrangian intersection pairing connect-ing x ⊗ a to y ⊗ b (left) can be pushed and collapsed (as indicatedby the arrows) to give a disk in the bordered Heegaard diagramconnecting x to y with compatible Reeb chords (right).1.2. Applications.
In this paper we apply Theorem 1.2 to study a question inknot concordance due to Hedden.
Conjecture 1.5 ([7]) . The only homomorphisms on the knot concordance groupsinduced by satellite operators are the zero map, the identity, and the involutioncoming from orientation reversal.In this paper we only consider Hedden’s conjecture in the smooth category, butwe remark it is open in both the topological and smooth category.Note if a pattern P induces a group homomorphism, then it must be a slicepattern (i.e. P ( U ) is a slice knot, where U is the unknot). In this paper, we restrictour attention to unknot patterns (i.e. P ( U ) = U ) that admit genus-one doubly-pointed Heegaard diagram. Note such patterns cannot be dealt with using theobrstruction coming from the Casson-Gordon invariant due to Miller [11].Our first step classifies all (1 , Theorem 1.6.
Unknot patterns admitting genus-one doubly-pointed bordered Hee-gaard diagrams are in one-to-one correspondence with patterns determined by two-bridge links.
We actually prove stronger results which imply Theorem 1.6: in Theorem 5.1 weclassify all the genus-one Heegaard diagrams that give rise to unknot patterns, and
NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , in Theorem 5.4 we give precise correspondence between such Heegaard diagramsand patterns determined by two-bridge links.Second, we give a formula for τ ( P ( T , )) and τ ( P ( − T , )), where P is any patterndetermined by a two-bridge link. Recall every two-bridge link admits a Schubertnormal form parametrized by a pair of coprime integers ( p, q ) such that p is even;denote such a link by b ( p, q ). Theorem 1.7.
Let P be a pattern knot obtained from a two-bridge link b ( p, q ) such that q > . Let w ( p, q ) be the winding number of P , and define σ ( a, b ) = P a − i =1 ( − ⌊ iba ⌋ . Then τ ( P ( T , )) = max( | w ( p, q ) | + σ ( p − q, q )2 + 1 , , and τ ( P ( − T , )) = min( −| w ( p, q ) | + σ ( p − q, q )2 , . Remark . Theorem 1.7 is of independent interest. When restricting the com-panion knot to the trefoil or the left-handed trefoil, Theorem 1.7 unifies previousresults on τ -invariant of satellite knots: the ( n, b (2 n, b (8 , b (14 ,
5) [5][6][8][9]. In fact, when fixed to an aforementioned specific pattern knot,one can reprove the satellite formula using the technique in this paper.The assumption that a pattern P induces a homomorphism on the concor-dance group constrains the behavior of the τ -invariant under the action by P , i.e. τ ( P ( K )) = | w ( P ) | τ ( K ) for any knot K (see the proof of Corollary 1.9 in Section 6or Proposition 5.4 of [11]). This together with Theorem 1.7 implies Corollary 1.9.
Let P be a pattern knot obtained from a two-bridge link b ( p, q ) such that q > . If | w ( p, q ) | 6 = 1 or σ ( p, q ) = − , then P does not induce a grouphomorphism on the smooth knot concordance group. Note that Hedden’s conjecture in particular implies any pattern with windingnumber (modulo sign) greater than or equal to two does not induce homomorphism.Corollary 1.9 confirms this within patterns obtained from two-bridge links. Weactually wonder if the behavior of the τ -invariant could completely answer thisquestion. Namely, Question 1.10.
Is there a pattern P with winding number | w ( P ) | ≥ τ ( P ( K )) = | w ( P ) | τ ( K ) for any knot K ?For slice patterns of winding number one, more subtle conditions are required toexclude the case in which the pattern is concordant (within the solid torus) to thecore; the σ -function in Corollary 1.9 is such a condition for patterns obtained fromtwo-bridge links. Actually, examining examples with | w | = 1 and σ = − Question 1.11.
Let P be a pattern determined by a two-bridge link b ( p, q ) with q >
0. If | w ( P ) | = 1 and σ ( p − q, q ) = −
1, then is P always concordant to the coreof the solid torus? WENZHAO CHEN
Finally, it might also be worth mentioning that it is unknown whether slicepatterns of winding number one always act as the identity map in the topologi-cal category, and hence presumably it is hard to study this case by concordanceinvariants that are blind to the smooth-topological difference.1.3.
Immersed train tracks for general pattern knots.
It is natural to expecta similar immersed curve interpretation can be extended to include general patternknots. At the course of writting this paper, it is not completely clear to the authorhow this can be achieved. Without genus-one bordered Heegaard diagrams, analgorithm must be given to translate the corresponding filtered bordered invariantto immersed train track on the torus. Such filtered invariant are often not reducedif we forget the filtration; this in particular prevents a direct application of thealgorithm given in [2].One might possibly achieve this by using the bimodule point of view discussedin [4]: \ CF DD -bimodules are expected to correspond to immersed surfaces (La-grangians) in T × T , and pairing with a [ CF A whose immersed curve is γ can beinterpreted as intersecting the surface with γ × T and then projecting it to thesecond T . View a pattern knot equivalently as a two-component link. If one cansuccessfully represent the \ CF DD -bimodule of this link complement as an immersedsurface, then pairing the surface with the doubly-pointed Heegaard diagram for thepattern corresponding to the Hopf link would produce an immersed curve for thepattern knot.In Section 7 we propose an approach in line with the spirit of working withfiltered object: we introduce a notion called filtered extendability , and give a wayto produce immersed train tracks for filtered extended type D structures. Filteredextendability is an analogue of the extendability condition appeared in [2], andis automatically satisfied by all type D structures arised from pattern knots. Inpractice, the approach gets us the desired immersed curves and pairing. We hencespeculate a favorable theory exists in general.
Organization.
Section 2 contains preliminaries on bordered Heegaard Floer ho-mology needed for the proof of Theorem 1.2, which is given in Section 3. Section4 contains a diagrammatic approach to compute the τ -invariant. In Section 5 weprove Theorem 1.6. In Section 6 we prove Theorem 1.7 and Corollary 1.9. Section7 contains a brief discussion for immersed curves for general patterns. Acknowledgments.
This project started when the author was a graduate studentat Michigan State University and was partially supported by his advisor MattHedden’s NSF grant DMS-1709016. The author would like to thank Matt Heddenfor his enormous help. The author would also like to thank Abhishek Mallick forinforming him of Ording’s result used in this paper. The author is grateful to theMax Planck Institute for Mathematics in Bonn for its hospitality and support.2.
Preliminaries
This section collects the relevant aspects of bordered Floer homology for 3-manifolds with torus boundary as a preparation for the proof of Theorem 1.2. Theexperts may well skip this section. For the readers who would like to see moredetail and other aspects of the theory, the author would recommend [2, 3, 8, 10].
NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , We outline the subsections: Section 2.1 recalls the definitions of a type D struc-ture and a type A structure, the box tensor product, and how to define the filtered [ CF A in terms of a genus-one doubly-pointed bordered Heegaard diagram; Section2.2 explains how to interpret \ CF D as train tracks on the torus; Section 2.3 explainsgradings of the bordered Floer package.2.1.
Bordered Floer homology.
We focus on bordered manifolds with torusboundary. To such manifolds, Lipshitz, Oszv´ath, and Thurston associated a typeD structure and a type A structure over the torus algebra up to certain quasi-ismorphism.The torus algebra A is given by the path algebra of the quiver shown in Figure3. As a vector space over F , A has a basis consisting of two idempotent elements ι and ι , and six “Reeb” elements: ρ , ρ , ρ , ρ = ρ ρ , ρ = ρ ρ , ρ = ρ ρ ρ . Figure 3.
The path algebra of this quiver with the specified re-lation is the torus algebra.Denote by I = h ι i ⊕ h ι i ⊂ A the ring of idempotents. A type D structure over A is a unital left I -module N equipped with an I -linear map δ : N → A ⊗ I N satisfying the compatibility condition( µ ⊗ I ) ◦ ( I ⊗ δ ) ◦ δ = 0 . Let δ = δ , and for k = 2 , , , ... , inductively define maps δ k = ( I ⊗ k − ⊗ δ ) ◦ δ k − .A type D structure is bounded if δ k = 0 for all sufficiently large k ; in this paper wewill always work with bounded type D structures.A type A structure is a right unital I -module M with a family of maps m i +1 : M ⊗ A ⊗ i → M , i ≥ n X i =1 m n − i ( m i ( x ⊗ a ⊗· · ·⊗ a i − ) ⊗· · ·⊗ a n − )+ n − X i =1 m n − ( x ⊗· · ·⊗ a i a i +1 ⊗· · · a n )and m ( x,
1) = xm i ( x, · · · , , · · · ) = 0 , i >
2A type A structure M and a type D structure N can be paired up to give achain complex via the box tensor product M ⊠ N . As a F vector space, M ⊠ N isisomorphic to M ⊗ I N . The differential is defined by ∂ ( x ⊗ y ) = ∞ X i =0 ( m i +1 ⊗ I N )( x ⊗ δ i ( y ))Requiring the type D structure to be bounded guarantees the sum in the aboveequation is finite, and hence the box tensor product is well-defined. WENZHAO CHEN
Given two 3-manifolds Y and Y with torus boundary, let h : T → ∂ ( Y )and h : − T → ∂Y be diffeomorphisms parametrizing the boundaries, and let Y = Y ∪ h ◦ h − Y be the glued-up manifold. Lipshitz, Ozsv´ath, and Thurstonassociated to ( Y , h ) a type A structure [ CF A ( Y ) and ( Y , h ) a type D structure \ CF D ( Y ), and showed that the box tensor product [ CF A ( Y ) ⊠ \ CF D ( Y ) is homo-topy equivalent to d CF ( Y ). In the case when there is a knot K ⊂ Y such that theinduced knot in the glued-up manifold Y is null-homologous, then one can associateto K ⊂ Y a filtered type A structure [ CF A ( Y , K ) so that the box tensor product [ CF A ( Y , K ) ⊠ \ CF D ( Y ) is homotopy equivalent to \ CF K ( Y, K ).All of the aforementioned objects are defined in terms bordered Heegaard dia-grams and involve counting certain J-holomorphic curves. For our purpose, belowwe only recall the definition of [ CF A ( Y , K ) when the bordered Heegaard diagramis of genus one. In this case, one could avoid (hide) the J-holomorphic curve theory.A genus-one doubly-pointed bordered Heegaard diagram D is a 5-tuple (Σ , α a , β, w, z )such that • Σ is a compact, oriented surface of genus one with a single boundary com-ponent. • α a consists of a pair of arcs ( a a , a a ) properly embedded in Σ, such that a a ∩ a a = ∅ and the ends of a a and a a appears alternatively on ∂ Σ • β is an embedded closed curve in the interior Σ such that Σ \ β is connected,and intersects α a transversely. • A basepoint w on ∂ Σ \ ∂α a , and a basepoint z in Int(Σ) \ ( α a ∪ β ), whereInt(Σ) denotes the interior of Σ. • Label the arcs on ∂ Σ, so that ( ∂ Σ , α a , w ) is as shown in Figure 4. We usethe symbol I ∈ { , , } to denote the arc obtained by concatenationof the arcs labeled by 1, 2, 3 accordingly. Figure 4.
Pointed match circleSuch a diagram specifies a 3-manifold with torus boundary and an oriented knot.The 3-manifold is obtained by attaching a 3-dimensional 2-handle to Σ × [0 ,
1] along β × { } , and the knot is the union of two arcs on Σ: on connects z to w in thecomplement of β , the other connects w to z in the complement of α a . We do notexplain, but simply point out the data also specifies a parametrization of the torusboundary. If a knot K in a bordered 3-manifold Y can be represented by a genus-1bordered Heegaard diagram, then we define a filtered type A module [ CF A ( Y , K )as following: NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , (1) [ CF A ( Y , K ) is generated by the set G = { x | x ∈ β ∩ α a } as a F vectorspace.(2) For x ∈ G , I acts on it as following x · ι = ( x if x ∈ α a ∩ β x · ι = ( x if x ∈ α a ∩ β I -action on [ CF A ( Y , K ).(3) View Σ = T − Int( B ), where B is a disk. Let ˜Σ be the covering space ofΣ which is obtained from the universal cover C of T by removing the liftsof B . For the maps m n +1 : M ⊗ A ⊗ n → M , n ≥
0, first define m n +1 ( x, ρ i , · · · , ρ i n ) = X y ∈G M ( x, y ) y. Here i j ∈ { , , , , , } for j = 1 , · · · , n , and M ( x, y ) is the count(modulo 2) of index 1 embedded disks in ˜Σ, such that when we traverse theboundary of such a disk with the induced orientation, we may start froma lift of x , traverse along an arc on (some lift of) α a , then along the arc i on (some lift of) ∂B , · · · , along the arc i n , along an arc on α a to a lift of y , finally it traverse along an arc on some lift of β from y to x . Also define m ( x,
1) = xm i ( x, · · · , , · · · ) = 0 , i > . The above three equations determine m n +1 .(4) Each term in m i has a relative Alexander grading difference that will spec-ified in Subsection 2.2.2.2. Gradings of the bordered Floer package.
The bordered Floer invariantsare graded by certain (coset spaces of) non-commutative groups. When restrictedto manifolds with torus boundary, the relevant grading group G is defined to be G = h ( m ; i, j ) | m, i, j ∈ Z , i + j ∈ Z i with the group law( m , i , j ) · ( m , i , j ) = ( m + m + 12 (cid:12)(cid:12)(cid:12)(cid:12) i j i j (cid:12)(cid:12)(cid:12)(cid:12) ; i + i , j + j )Here m is called the Maslov component , and ( i, j ) is called the spin c component .In the presence of a knot we would also like to record the Alexander grading ofthe corresponding invariants, and this leads to using the enhanced grading group e G = G × Z . The new Z summand is called the Alexander factor .There are two elements in e G will be relevant to us λ = (1; 0 ,
0; 0) µ = (0; 0 , − The torus algebra A is graded by e G by setting gr ( ι i ) = (0; 0 ,
0; 0) , i = 1 , gr ( ρ ) = ( −
12 ; 12 , −
12 ; 0) gr ( ρ ) = ( −
12 ; 12 ,
12 ; 0) gr ( ρ ) = ( −
12 ; − ,
12 ; 0)and require gr ( ρ I · ρ J ) = gr ( ρ I ) gr ( ρ J ), for I, J ∈ { , , , , , } .A type D structure \ CF D ( Y ) decomposes as direct sum over spin c -structures of Y . Fixing a spin c structure s , the corresponding component is graded by certainright coset space of e G gr : \ CF D ( Y , s ) → e G/ ∼ The definition of this grading function involves utilizing some concrete Heegaarddiagram and is not necessary for our purpose. Instead, we recall gr satisfies gr ( δ ( x )) = λ − gr ( x ) and gr ( ρ I ⊗ x ) = gr ( ρ I ) · gr ( x ).Similarly, a filtered type A structure decomposes as direct sum over spin c -structures of Y . Fixing a spin c structure s , the corresponding component is gradedby certain left coset space of e Ggr : [ CF A ( Y , K, s ) →∼ \ e G The property that will be relevant to us is if B is a domain connecting x to y , then gr ( x ) gr ( B ) = gr ( y ), where gr ( B ) = ( − e ( B ) − n x ( B ) − n y ( B ); gr ( ∂ ∂ B ); n w ( B ) − n z ( B )) . Here e ( B ) is the Euler measure , ∂ ∂ B is the sequence ( ± ρ i , · · · ± ρ i k ) of Reebchords appearing at the boundary of B (where the sign indicates the orientation),and gr ( ∂ ∂ B ) refers to the spin c component of gr ( ρ i ) ± · · · gr ( ρ i k ) ± . The gradings on the type D and type A structure induces a grading on the boxtensor product gr : [ CF A ( Y , K, s ) ⊠ \ CF D ( Y , s ) →∼ \ e G/ ∼ x ⊗ y gr ( x ) gr ( y )Two elements x ⊗ y , x ⊗ y ∈ [ CF A ( Y , K, s ) ⊠ \ CF D ( Y , s ) corresponds to thesame spin c structure if and if there exist M and A such that gr ( x ⊗ y ) = gr ( x ⊗ y ) λ M µ A . In the case, M is the Maslov grading difference: M ( x ⊗ y ) − M ( x ⊗ y ),and A is the Alexander grading difference: A ( x ⊗ y ) − A ( x ⊗ y ).2.3. Type D structure and immersed train tracks.
We recall how to representa type D structure as immersed train tracks in a punctured torus.A type D structure N can be represented by an decorated graph. Let N i = ι i · N , i = 0 ,
1, then N = N ⊕ N . Let B i be a basis for N i , i = 1 ,
2. Fixing such a basis B = B ∪ B for N , we construct a decorated graph Γ. The vertices are in one-to-onecorrespondence with elements in B , and we label a vertex with • if it correspondsto an element in B , otherwise we label it with ◦ . For two vertices corresponding tobasis elements x and y , we put a directed edge from x to y labeled with I whenever ρ I ⊗ y is a summand in δ ( x ), where I ∈ {∅ , , , , , , } . We call a decoratedgraph is reduced if none of the edges is labeled by ∅ . NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , A reduced decorated graph can equivalently be represented by an immersedtrain track in the punctured torus (
T, w ). More specifically, let T = R / Z , let w = (1 − ǫ, − ǫ ), and let α and β be the image of the y and x -axes. To constructthe immersed train track, embed the vertices of the decorated graph into T so thatthe • vertices lie on α in the interval 0 × [ , ], and the ◦ vertices lie on β in theinterval [ , ] ×
0, and then embed the edges in T according to its label accordingto the rule as shown in Figure 5. We require the edges to intersect transversely. Figure 5.
Basic rules for representing the type D structures astrain tracks.In general, the train tracks thus obtained are usually not immersed curves. Workin [2] shows that for type D structures arised from 3-manifolds with torus boundary,one can always pick some particularly nice basis, so that the train tracks obtainedfrom the corresponding decorated graphs are immersed curves (possibly decoratedby local systems). 3.
Proof of the main theorem
Proof of Theorem 1.2.
Let µ K and λ K be the meridian and longitude of the com-panion knot K . Let α ( K ) be a train track representing \ CF D ( X K , µ, λ ). Forconvenience, we assume α ( K ) comes from a restriction of d HF ( X K ). ( d HF ( X K )represents an exented type D structure, and α ( K ) is the curve-like sub-diagramof d HF ( X K ) representing the underlying type D structure.) Let H =(Σ , α a , β, w, z )be a genus-one bordered diagram for ( S × D , P ) corresponding to the standardmeridian-longitude parametrization of the torus boundary. We first place α ( K ) and( β, w, z ) in a specific position on the torus T : Identify T as the obvious quotientspace of the squre [0 , × [0 ,
1] and divide the square into four quadrants by the seg-ments { } × [0 ,
1] and [0 , × { } . Include α ( K ) into the first quadrant and extendit horizontally/vertically. Include H into the third quadrant so that w is placednear (0 ,
0) and α a is on the boundary of the third quadrant, then forget Σ and α a ,and extend β horizontally/vertically. See Figure 6. For the ease of notation, we set α = α ( K ).We claim it suffices to prove(3.1) \ CF K ( T , α, β, w, z ) ∼ = [ CF A ( P, w, z ) ⊠ \ CF D ( X K , µ K , λ K )To see the claim, first note the above placement of α and β can be viewed as theresult applying a specific representative of the homeomorphism h as stated in thetheorem. Secondly, regular homotopies of curve-like train tracks do not change theLagrangian intersection Floer homology (Lemma 35 of [2]).It is easy to see the isomorphism in 3.1 as vector spaces: The intersections of α and β only occur in the second and fourth quadrants, and those in the second quad-rant are in one-to-one correspondence with the tensor product of the ι components Figure 6.
Putting α ( K ) and β ( P ) in a specific position.of [ CF A ( P, w, z ) and \ CF D ( X K , µ K , λ K ), while those in the fourth correspond tothe tensor products of the ι components.We move to analyze the differentials. For convenience, we work with the universalcover π : R → T of T . Let ˜ β be a connected component of the π − ( β ). If α is connected, we let ˜ α be a connected component of the π − ( α ). Otherwise, let α ′ be a lift of α to R , and let ˜ α = ∪ i ∈ Z t i ( α ′ ), where t is the horizontal coveringtranslation. Note \ CF K (˜ α, ˜ β, π − ( w ) , π − ( z )) = \ CF K ( α, β, w, z ).Let x = x ⊗ x and y = y ⊗ y be two intersection points of α and β . Givena holomorphic disk connecting some lift ˜ x and ˜ y of x and y that contributes tothe differential of \ CF K (˜ α, ˜ β ), we claim there is a corresponding matching typeD and type A operation giving the desired differential via the box-tensor productoperation.To prove the claim, we first explain how a holomorphic disk as above induces atype A operation in [ CF A ( P, w, z ). To do this, we define a collapsing operation on R that sends holomophic disks in R to holomorphic disks in Σ. The collapsingoperation is defined to be the composition of the following five operations (seeFigure 7):(Step 1) Assume the w -base point has coordinate [ ǫ, ǫ ] on [0 , × [0 , √ ǫ around every integer point in R .(Step 2) Enlarge the holes to a hook shaped region by pushing the boundary, andwhile doing so, let ˜ α move along accordingly. NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , Figure 7.
Five steps in the collapsing operation.(Step 3) Enlarge the holes one more time to a rounded square by pushing the bound-ary.(Step 4) Collapse the lifts of the second quadrant.(Step 5) Collapse the lifts of the fourth quadrant.We choose the enlargement in Step 2 and 3 so that after Step 4 and 5, eachhollowed region is bounded by a circle of radius √ ǫ . Denote the resulting space ofthe collapsing operation by ˜Σ.Note ˜Σ can be identified with a covering space of the bordered diagram (Σ , α a , β, w, z ):the circles are the lifts of the pointed circle ( ∂ Σ , w ), ˜ β is sent to a curve that covers β , and the horizontal and vertical segments connecting the circles are lifts of α a .The arcs on ˜ α are sent to arcs on ˜Σ that traverse along lifts of α a and boundarycircles according to the following rule: Figure 8.
Degeneraitions(1) For I ∈ { , , , , , } , a ρ I -arc in the lifts of the first quadrant is sentto ρ I on the pointed match circle.(2) Arcs in the lifts of the second/fourth quadrant are projected horizontally/verticallyto the lifts of the α a .Let φ : D → R be a holomorphic disk connecting some lift of x = x ⊗ x and˜ y = y ⊗ y , further assume φ does not cross the w -base points and has Maslovindex one. We will write D φ for the domain of φ , and use them interchangeably byabusing notations. Let D A be the image of D φ under the collapsing operation, thisis a domain connecting x and y . We claim D A determines an embedded disk in ˜Σ,and the Reeb chords appearing on ∂D A is given by ∂ ˜ α φ . Assuming this claim, we seethe differential induced by φ has a correspondent in the box tensor product. Since D A is an embedded disk, it gives a type A operation in [ CF A ( P, w, z ). On the otherhand, D A being embedded implies with the induced orientation, the Reeb chordson ∂D A are oriented consistently with the convention to produce the immersedcurves of a type D module. As ∂ ˜ α D φ gives the same sequence of Reeb chords as ∂D A , the type D operation in \ CF D ( X K , µ K , λ K ) determined by ∂ ˜ α D φ pairs withthe type A operation induced by D A . So a differential in d CF ( α, β ) corresponds toa differential in [ CF A ( P, w, z ) ⊠ \ CF D ( X K , µ K , λ K ).We move to prove the aforementioned claim. Suppose the claim is not true, inview of the construction of the collapsing operation, then part of ∂ ˜ α φ are pinchedtogether during the collapsing operation, creating needle and bubble degenerationas shown in Figure 8. So it suffices to prove such degeneration do not happen. Tosee needle degeneration do not exist, simply note that the tip of a needle wouldcorrespond to an ∅ -arrow in \ CF D ( X K , µ, λ ), which contradicts to our assumptionthat the type- D module is reduced. To see bubble degeneration do not exist, weseparate the discussion into two steps. First, we observe that there are no bubbledegeneration bounding disks. If not, as the boundary of the bubble consists of α -arcs and Reeb chords, the disk it bounds must contain some lifts of the w -basepoint; this would imply D φ also contains the w -base point and hence contradictsto our assumption. Secondly, as we may now assume no bubbles bound disks, theexistence of bubbles would imply there is a “hollowed” bubble: if we orient thebubble counterclockwise, the region D A appears on its left (see Figure 8), yet sucha bubble again implies that D A contains the w -base point: the boundary of the“hollowed” bubble corresponds to a loop in ˜Σ that consists of α -arcs and Reeb NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , Figure 9.
Degeneraitionschords that do not cross any lifts of the w -base point, and any outward region abutit must contain some lift of the w -base point (as shown in Figure 9). This finishesthe proof of the claim.Conversely, suppose there is a differential in [ CF A ( P, w, z ) ⊠ \ CF D ( X K , µ K , λ K ),then we know there is a disk D A in ˜Σ corresponding to the type A operation,and there is a curve in \ CF D ( X K , µ, λ ) corresponding to the type D operation.Inverting the collapsing operation, we see there is a holomorphic disk giving thedesired differential in \ CF K ( α, β ).Now we move to prove the isomorphism also preserves the relative Alexandergrading. Symmetry of the knot Floer homology would then imply the absoluteAlexander grading are also preserved. From now on, we will assume α is connectedfor convenience.Recall from the preliminaries that bordered Floer invariants are graded by certaincoset spaces of the enhanced grading group e G . For considering the Alexandergrading, it suffices to use a simpler group G A , which is obtained from e G by forgettingthe Maslov component. Abusing the notation, we will still denote the gradingfunction by gr , even though its value are now in (coset spaces of) of G A .Given x = x ⊗ x , y = y ⊗ y ∈ [ CF A ( P ) ⊠ \ CF D ( X K ), let ˜ x = ˜ x ⊗ ˜ x and˜ y = ˜ y ⊗ ˜ y be the corresponding lifts. Let ˜ P be a path on ˜ β connecting ˜ y to ˜ x ,and ˜ P be a path on ˜ α connecting ˜ x to ˜ y . Then ˜ P ∪ ˜ P bounds a domain ˜ B in R . Under the covering projection, ˜ B gives rise a domain B ′ in T ; by subtractingor adding copies of T , we may assume B ′ does not contain the w -base point. Notewe can also perform the collapsing operation on T to get to Σ, and this gives riseto a domain B ⊂ Σ. Denote by ρ ( ˜ P ) the sequence of Reeb chords determinedby ˜ P (the order is induced by the orientation of ˜ P ). Note gr ( ∂ ∂ B ) = gr ( ρ ( ˜ P )),and gr ( y ) = gr ( ρ ( ˜ P )) − gr ( x ), and gr ( y ) = µ n z ( B ) − n w ( B ) gr ( x ) gr ( ∂ ∂ B ) (recall µ = (0 , − gr ( y ⊗ y ) = µ n z ( B ) − n w ( B ) gr ( x ) gr ( ∂ ∂ B ) gr ( ρ ( ˜ P )) − gr ( x )= µ n z ( B ) − n w ( B ) gr ( x ⊗ x )Finally, note n z ( ˜ B ) − n w ( ˜ B ) = n z ( B ) − n w ( B ). Therefore, the relative Alexan-der grading induced by pairing the bordered Floer invariants equals that of theLagrangian Floer chain complex \ CF K ( α, β, w, z ).For the Maslov grading, one only needs to modify the argument in Section 2.3and 2.4 of [3]. Here we point out the extra cares needed to be taken in this case,and refer the reader to [3] for details. Note the curve β ⊂ ∂ ( S × D ) is actuallythe immersed curve corresponding to a subdiagram of the decorated diagram for \ CF D ( P, w, z ). The argument in [3] deals with the Maslov grading in the casewhen pairing two reduced bordered invariants. The extra care needed for the non-reduced case is understanding the effect of ∅ -arrows: when dealing with such arrows,we degenerate it into a folded line segment, hence both the area contribution andadjusted area contribution would be zero, and the adjusted path contribution is −
1. With this at mind, the argument in [3] can be adapted to the current setting.It is worth pointing out the immersed curves, as Lagrangian, are in general notembedded and sometimes are even obstructed. Therefore, we recall the definitionof Maslov grading difference used in [3] for the convenience of computation.
Definition 3.1.
Let γ and γ be immersed train tracks in T , and let x, y ∈ γ ∩ γ .Suppose there is a path p i from x to y on γ i , i = 0 ,
1, such that p − p lifts to aclosed loop l in R \ Z . The Maslov grading difference m ( y ) − m ( x ) is twice thenumber of lattice points enclosed by l (where each point is counted with multiplicitythe winding number of l ) plus π times the net total rightward rotation along thesmooth segments of l . (cid:3) Computing the τ -invariant of satellite knots In this section, we give a way to compute the τ -invariant by manipulating thepairing diagram for \ CF K ( P ( K )) when P is a (1 , τ -invariant can be defined as the Alexandergrading of the cycle surviving the infinity page. Each time when we pass fromone page to the next, it amounts to cancel differentials that connects elements ofthe minimal Alexander filtration difference. Such cancellations can be done in thediagram: continuing with the notation in Theorem 1.2, one can isotope the curve β to eliminate pairs of intersection points of minimal filtration difference by pushingthe curve across embedded Whitney disks. At the end of such isotopies, only oneintersection point is left, which corresponds to the cycle that survives the infinitypage. Hence the Alexander grading of this last intersection point is the τ -invariantof the satellite knot.In the practice of carrying out this procedure we need to remember the filtrationdifference of the intersection points. To do this, we introduce the so called A-buoys ,which will be arrows attached to the β curve. To explain how the A-buoys work,we first give the following lemma. NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , Figure 10.
Computing the τ -invariant for M ( T , ): x is the lastintersection point left and τ ( M ( T , )) = A ( x ) = 2 . Lemma 4.1.
Using the notation in Theorem 1.2, let x , y be two intersection pointsof α and β . Let l be an arc on β from x to y , and let δ w,z be a straight arc connecting w to z . Then A ( y ) − A ( x ) = l · δ w,z .Proof of Lemma 4.1. Let D be a Whitney disk connecting x to y such that ∂ β D = − l . Then A ( y ) − A ( x ) = n w ( D ) − n z ( D ) = − ∂D · δ w,z . Note ∂ α D · δ w,z = 0 (Figure6) and hence the lemma follows. (cid:3) With this lemma understood, note if we perform an isotopy of β by pushing itacross an embedded Whitney disk that contains the z -base point, the (algebraic)intersections of β with δ w,z are changed. To remedy, we put a right amount of smallarrows on β whenever such an isotopy is performed, and then when we count theAlexander filtration difference, we count both the algebraic intersection between thecorresponding arc on β an δ w,z , together with intersections of this arc and thesenewly added small arrows. These small arrows are the so-called A-buoys. Example 4.2.
We continue considering the satellite knot M ( T , ), where M is theMazur pattern. Note we first cancel some differentials whose filtration differenceare one (Left of Figure 10), and then we are left with five intersection points x i , i = 1 , · · · , x and x is two as indicated by the A-buoys, while x and x , x and x has filtrationdifference one, coming from the intersection with δ w,z . So we cancel the differentialsof filtration difference one first, leaving x as the remaining intersection point. Oncewe figure out the relative Alexander grading of \ HF K ( M ( T , )) using the diagramon the left of Figure 10, we use the symmetry of the knot Floer homology we can alsodetermine the absolute Alexander grading x . Finally τ ( M ( T , )) = A ( x ) = 2 . The classification of (1 , -unknot patterns We call a pattern P to be an unknot pattern if the satellite knot constructed byapplying P to the unknot is isotopic to the unknot. Knot Floer homology detectsthe unknot: a knot K is the unknot if and only if \ HF K ( K ) ∼ = F [13]. Exploitingthis fact and Theorem 1.2 we give a classification of (1 ,
1) unknot patterns (i.e.unknot patterns that admit a genus-one Heegaard diagram).
We first classify all the genus-one Heegaard diagrams that give rise to unknotpatterns.
Theorem 5.1.
Nontrivial genus-one doubly-pointed bordered Heegaard diagramsthat give rise to unknot patterns are in one-to-one correspondence with pairs ofintegers ( r, s ) such that | r | ≥ , and gcd(2 | r | − , | s | + 1) = 1 . Figure 11.
The 5-tuple from a bordered Heegaard diagram
Proof.
Recall a genus-one Heegaard diagram (Σ , α a , β, w, z ) is equivalent to a 5-tuple ( β, µ, λ, w, z ) ⊂ T (Figure 11). So to classify bordered diagrams for unknotpatterns, we equivalently classify their corresponding 5-tuples. Lemma 5.2.
Let ( β, µ, λ, w, z ) ⊂ T be a 5-tuple constructed from a genus-oneHeegaard diagram, then β · µ = 0 and β · λ = ± , where we arbitrarily orient thecurves. In particular, if the -tuple gives rise to an unknot pattern, then up toisotopy, λ and β has a single intersection point.Proof of Lemma 5.2. Note if we ignore the z -base point, then the doubly pointedHeegaard diagram descends to a bordered diagram for the solid torus with standardparametrization of the boundary. Hence if we remove z in the 5-tuple, up to isotopyit is represented by the diagram shown in Figure 12. This implies β · µ = 0, and β · λ = ±
1. Now assume the 5-tuple is constructed from an unknot pattern P . Figure 12.
Bordered diagram for the solid torusPair this diagram with the immersed curve associated to the unknot complement(which is a horizontal line parallel to λ ) using Theorem 1.2. Then \ HF K ( P ( U )) ∼ = F NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , implies up to isotopy in the complement of w and z , β can be arranged to intersect λ geometrically once. (cid:3) Figure 13.
Ten types of fundamental regions of bordered Hee-gaard diagrams for unknot patternsUp to isotopy, assume µ and λ intersect geometrically at one point, cut T openalong µ , λ , and call the resulting region the fundamental region . Then Lemma 5.2implies, up to isotopy of β in the complement of w and z , the diagram of the 5-tuplein the fundamental region is one of the ten types as shown in Figure 13.We further classify the last eight nontrivial types. Note that the last four typesare vertical reflections of the (AR), (BR), (AL), and (BL), so it suffices to classify (AR), (BR), (AL), and (BL). Further note that (AR) and (BR) are horizontalreflections of (AL) and (BL), so it suffices to classify (AL) and (BL).Note in all the nontrivial cases, the diagram is determined by a pair of numbers:the number of loops around w and z , and the number of strands in the middle stripethat separates w and z . This is because the rest of the arcs are determined by thecondition β · µ = 0. Yet simply parametrizing each cases by this pair of numbersis not very convenient. Instead, we further group (AR) and (BR) together, call ittype (R). Similarly, the other cases are grouped in pair to give (L), (MR), (ML). Figure 14.
Converting (AL) to (BL). The main regions are boxed.By an isotopy, the main region of an (AL) diagram can be con-verted into one of a (BL) diagram.We explain why we group the pairs together using the example of (AL) and(BL). Observe that we may do an isotopy of (AL) as in Figure 14, so that the regioncontaining the loops and the middle stripe is the same as the corresponding region incase (BL). We call this region the main region . Note that by the condition β · µ = 0,the number of loops r and strands s in the stripe in the main region determinesthe rest of the diagram. In particular, the pair ( r, s ) will determine whether theresulting diagram is of type (AL) or (BL). In summary, type (L) diagrams are inone-to-one correspondence with pairs of non-negative integers ( r, s ) such that r ≥ β -curve has a single connected component.We characterize the pairs ( r, s ) whose resulting β -curve has a single connectedcomponent. Lemma 5.3.
Given a pair of non-negative integers ( r, s ) such that r ≥ , thenthe resulting β -curve in the main region of a type (BL) diagram with r loops and s bridges has a single connected component if and only if gcd(2 r − , s + 1) = 1 .Proof of Lemma 5.3. Label the intersection points of β and µ in the main regionby 0 , · · · , r + s (Figure 15). Let a = 0, traverse a connected component of β inmain region starting from the out-most loop around w , and denote the intersectionpoints with µ by a , · · · , a n . Note there is a sequence of numbers { ǫ i } ∈ {± } suchthat ǫ i +1 a i +1 − ǫ i a i ≡ r − s + 2 r )) . In fact, we may take ǫ i to be sign of the intersection of β and µ at a i . Now β hasa single connected component if and only if the subgroup generated by 2 r − NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , Figure 15.
The main region of a type (BL) diagram Z s +2 r ) contains the element 0, 1 or −
1, ..., 2 r + s or − (2 r + s ). This means 2 r − Z s +2 r ) . Hence this is equivalent to gcd(2 r − , s + 2 r )) = 1, which issame as gcd(2 r − , s + 1) = 1. (cid:3) Therefore, each of the cases (L), (R), (ML), and (MR) are in one-to-one corre-spondence with such pairs ( r, s ). To uniformly parametrizes all four cases, we allow r and s to be both positive and negative. We set the signs of the parameters ( r, s )corresponding to cases (R), (L), (MR), and (ML) to be (+ , +), (+ , − ), ( − , + , ), and( − , − ) respectively.Finally, one can covert the diagram in the fundamental region to a borderedHeegaard diagram (See Figure 16for examples). This finishes the proof of Theorem5.1. (cid:3) Now we recognize the patterns from the doubly pointed diagrams. Eventually, wewill identify such patterns as those obtained from . All 2-bridge links(knots) admit a presentation called the
Schubert normal form . Such a normal formis parametrized by a pair of coprime integers p and q such that p >
0, 0 < | q | < p ,and is denoted by b ( p, q ). (See Figure 17 for an example.) In general, b ( p, q ) is themirror image of b ( p, − q ), b ( p, q ) is a two-component link if and only if p is even, and b ( p, q ), b ( p ′ , q ′ ) are isotopic to each other if and only if p = p ′ and q ′ ≡ q ± (mod p ). Theorem 5.4.
Let P be a (1 , -unknot pattern obtained from a genus-one doubly-pointed bordered diagram of parameter ( r, s ) . Then the link consists of P and themeridian of the solid torus is the 2-bridge link b (2 | s | + 4 | r | , ǫ ( r )(2 | r | − . Here ǫ ( r ) is the sign function of r .Proof. The pattern knot P can be drawn on the torus by an arc connecting w to z in the complement of the β -curve, and then an arc in the complement of α a .Viewing in the fundamental region, P has a diagram consisting of two bundles of | r | − | s | + 1 arcs (Figure 18). Such a P together with themeridian of the solid torus is the 2-bridge link b (2 | s | + 4 | r | , ǫ ( r )(2 | r | − (cid:3) Figure 16.
Converting diagrams of 5-tuple into bordered Hee-gaard diagrams. On the left, a type (AR) diagram and its bor-dered Heegaard diagram. On the right, a type (BL) diagram andits bordered Heegaard diagram.
Figure 17.
The 2-bridge link b (10 , τ -invariant and two-bridge patterns Recall from the previous section that a two-bridge link b ( p, q ) gives rise to a(1 , P . In this section, we derive a formula for τ ( P ( T , )) and τ ( P ( − T , )) in terms of p and q . The formula involves two functions; one is aquantity σ associated to some Heegaard diagram for such a pattern, and the otheris the winding number w ( P ). We explain these quantities below.First a remark on the convention, from the previous section we know bordereddiagrams for such patterns are parametrized by a pair of integers, and in thissection, we will restrict attentions to diagrams of type (L), given by pairs ( r, − s ) NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , Figure 18.
The pattern knot. Note such a pattern is determinedby the number of loops and strands in the middle stripe of thediagram.with r, s ∈ Z , r >
1, and gcd(2 r − , s + 1) = 1. Note generality is not lost by thisrestriction as we may pass to mirror image of P ( K ) when P is not of type (L).We explain how we associate a quantity σ to a genus-one Heegaard diagramcorresponding to an unknot pattern. Definition 6.1.
Given a type (L) diagram H = H ( r, − s ), σ ( H ) is defined to bethe algebraic intersection number of the arc on β outside of the main region and aleft push-off of meridian µ (still denoted by µ ). Here we orient β and µ compatiblyso that if we isotope µ to β (disregarding w and z ), then the orientation of β isinduced from that of µ (See Figure 19). Figure 19.
Seeing σ ( H ) diagramatically. Note the boxed regionis the main region.For convenience, we call the arc out of the main region the dependent arc , as itis determined by the number of caps r around the w - and z -base points and thenumber of stripes s in the main region. We also remark that it is necessary to usea push-off in the above definition so that the end points of the dependent arc is noton the push-off meridian, and actually a more precise definition can be given usinghomology class in some relative homology group, but here we would rather stickwith this more straightforward and pictorial explanation. Note depending on the parity and sign of σ , one can draw the 5-tuple diagramcorresponding to H in a symmetric way as shown in Figure 20. Figure 20.
Symmetric diagrams for H ( r, − s ) in the fundamental region.Next, we give a closed formula for σ . Proposition 6.2. σ ( H ( r, − s )) = σ ( s + 1 , r −
1) = r − X i =1 ( − ⌊ i (2 r − s +1 ⌋ Proof.
Note since β · µ = 0, we may equivalently study the arc on β inside the mainregion, which we denote by l . Note l · µ is can be described by a simple formula:(6.1) l · µ = 1 − r + s − X i =1 ( − ⌊ i (2 r − r + s ⌋ We explain where the terms in the above formula come from. First note there area total 2 r + s many intersection points between l and µ , which we label from 0 to2 r + s − l downwards, thefirst intersection point is positive, and hence contributes 1 to the right hand side ofEquation 6.1. As we move on, the next intersection is negative. Note consecutiveintersection points on l will differ by 2 r − r + s and the furtherintersections change sign only when we need to make a turn along some caps, whichis captured by the floor function. NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , Now β · µ = 0 implies σ ( s + 1 , r −
1) + 1 − P r + s − i =1 ( − ⌊ i (2 r − r + s ⌋ = 0. Therefore, σ ( s + 1 , r −
1) = r + s − X i =1 ( − ⌊ i (2 r − r + s ⌋ − r − X i =1 ( − ⌊ i (2 r − s +1 ⌋ , where the last equality is proved in Lemma 3.9 of [1]. (cid:3) We move to consider the winding number w ( P ). In order to remove ambiguityof sign when speaking of the winding number, firstly we fix the convention on theorientations of the relevant objects: we orient the meridian µ of the solid torusso that in the 5-tuple diagram, µ is oriented downwards, and the pattern knot P is oriented so that the short arc connecting w to z in the complement of α a isoriented from w to z (Figure 18). By pushing P in the interior of S × D , we set w ( P ) = lk ( P, µ ). Lemma 6.3.
Equip β with the orientation obtained by isotoping µ to β . Recall δ w,z denotes the straight arc connecting w to z within the fundamental domain ofthe 5-tuple. We have w ( P ) = β · δ w,z Proof.
Since β is isopotic to µ , lk ( P, β ) = lk ( P, µ ). Figure 18 shows we have adiagram of P and β such that the only intersections between P and the meridiandisk bounded by β occur on the arc obtained by pushing δ w,z into the solid torus.Therefore, β · δ w,z = lk ( β, P ) = w ( P ). (cid:3) Figure 21. (a) shows the case in which σ is even, and in the thisexample O w ( H ) = O z ( H ) = 2; (b) shows the case in which σ isodd, and in the this example O w ( H ) = O z ( H ) = 1 . H ( r, − s ), lift the β curve to the covering space S × R of T corresponding to the subgroup of π ( T ) generated by the longitude λ . Labelthe lifts of the base point w as in Figure 21. The rule is: if σ ( H ) is even, then thefirst w base point to the right of β ∩ λ is labeled by 1, if σ is odd, we label thecorresponding point by 1 .
5, and in both cases, the labels increase by 1 as we movefrom one base point to the next from left to right. Note the lift of the β curveseparates S × R into two regions,which we call by the left region and right region.We define O w ( H ) to equal to the label of the right-most w base point contained inthe left region. One can similarly define O z ( H ). See Figure 21 for an example. Proposition 6.4.
Let P be a pattern obtained from the bordered Heegaard diagram H ( r, − s ) , then w ( P ) = O w ( H ) + O z ( H ) = 2 O w ( H ) .Proof. It is clear O w ( H ) = O z ( H ) in view of the symmetry of the diagram. Henceit is left to show w ( P ) = O w ( H ) + O z ( H ). By Lemma 6.3, w ( P ) = β · δ w,z ,and we may equivalently consider this intersection in S × R : let ˜ β be a singlelift of β , and π − ( δ w,z ) be the preimage of δ w,z of the covering map π . Then β · δ w,z = ˜ β · π − ( δ w,z ). The proposition then follows from the following observation:Let a w,z be a connected component of π − ( δ w,z ), then ˜ β · a w,z = 0 if both end pointsof a w,z are contained in the left region or the right region. ˜ β · a w,z = 1 if the w endpoint of a w,z is in the left region, while the z end point is on the right. Otherwise,˜ β · a w,z = − (cid:3) Remark . We remark there is also a closed formula for the winding number ininterest of computation.For the pattern corresponding to the two-bridge link b ( p, q )where q >
0, the winding number is equal to(6.2) w ( p, q ) = ⌊ p − ⌋ X k =0 ( − ⌊ (2 k +1) q ) p ⌋ We skip the proof for this formula and remark that it is similar to the proof ofProposition 6.2.For the proof of Theorem 1.7, we will partially carry out the algorithm in Section4: Do isotopies that cancel pairs of intersection points whose Alexander gradingdifference is one, after that we can read off the τ -invariant from the pairing diagram.More concretely, we will push the caps around the z -base point off one by one, untilthis cannot be done any more (See Example 4.2). Proof of Theorem 1.7.
For the ease of exposition, we give details in the case whenthe σ ( p, q ) is odd, | σ ( p, q ) | ≥ T , . We remark thatsimilar reasoning would work in other cases.We begin with the case when σ is odd and σ ≥
3. Note when the companionknot is the trefoil knot, in the universal cover of the pairing diagram, only two rowscontains the intersection points. We refer to them as the upper row and the lowerrow respectively.First, we examine the isotopy of pushing the z-caps off in the lower row. Pushthe innermost z-cap off and cancel intersection pairs as many as possible, then the β curve could possibly end with one of the four cases as shown in Figure 22, whichwe call U , U , L , and L . Note U increase the algebraic intersection number ofthe dependent part of β with µ , while U decreases. Same observation applies to L and L . NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , Figure 22.
Pushing β along the grey Whitney disks can possiblyend up with four situations.Note the A-buoys occur in U , L , L are out of the main region (recall this isthe region containing the caps and the stripes in the middle), and hence will not getinvolved in the next round of the z-cap removal. The A-buoy which comes from U will neither get involved if further z-cap removals end with U , L , or L . The onlydifference is if another U happens, then it creates a Whitney disk which connectsa pair of points whose Alexander grading difference is two (see Figure 23). Figure 23.
Two endings with U creates a Whitney disk havingfiltration difference equal to two.Note in the process of repeatedly carrying out the z-cap removals, the effect of L and L cancel each other: one increases the algebraic intersection number betweenthe dependent part of β and µ while the other decreases, and the A-buoys wouldcome in different directions and hence offset each other. The same observationapply to U and U . Figure 24.
Six types of possible situations occur in the processof removing z-capsWith these observations at hand, note during the process of the doing the z-capremovals we will be seeing one of the following six situations in Figure 24; one cancheck if a further z-cap removal is done to one of the six diagrams, the resultingdiagram is still one of them. The process of removing z-caps will end with one ofthe four types of diagrams as shown in Figure 25.The same reasoning can be applied to the analyze the isotopy in the upper row.At the end of this process, the diagram would look like one of the four cases shownin Figure 26.We move to combine the diagrams from Figure 25 and Figure 26. Note if we allowisotopies of the β -curve that eliminates Whitney disks connecting intersection pairswith Alxander filtration greater than or equal to two, then the ending diagram inthe upper row should be isotopic to the one in the lower row. With this understood,one can see there are three type of possibilities of how the simplified pairing diagramlooks like Figure 27, Figure 28, and Figure 29. In these figures, the dotted capsstand for those having two A-buoys on the tip. From Figure 24, one can see suchcaps would appear in the lower row to ensure that the first turn as we traverse NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , Figure 25.
Possible endings in the lower row when σ ≥ Figure 26.
Possible endings in the upper row when σ ≥ β curve downwards would appear after it crosses µ from right to left atleast ⌊ σ ⌋ many times. Similar observation applies to the appearance of such capsin the upper row. We move to determine which intersection point has the Alexander grading thatequals to the τ -invariant case by case. Figure 27.
Combining Figure 26 (1) or (2) and Figure 25 (1) or (2)The case given in Figure 27 comes from combining Figure 26 (1) or (2) andFigure 25 (1) or (2). If the dotted cap does not appear, then τ = A ( x ), as x is an intersection point with neither incoming nor outgoing differentials. Ifthe dotted cap appears, the relevant component of the chain complex consists ofthree intersection points: x , x , and y . The differentials are x → y , x → y .Therefore, τ = max( A ( x ) , A ( x )) . In the latter case, we claim A ( x ) ≥ A ( x ).To see this, note A ( g ) − A ( x ) = − A ( g ′ ) − A ( g ) = β · δ w,z = − w ( P ), and A ( x ) − A ( g ′ ) ≤ w ( P ) ≥ O w ( H ) is not changed under this z-capremovals), A ( x ) ≤ A ( g ′ ) + 1 = A ( g ) − w ( P ) + 1 ≤ A ( x ). Therefore, τ = A ( x ) inthis case. Figure 28.
Combining Figure 26 (1) and Figure 25 (3) or (4)The case given in Figure 28 comes combining Figure 26 (1) and Figure 25 (3) or(4). Similarly we have τ = max( A ( x ) , A ( x )). Note in this case, A ( x ) − A ( x ) = β · δ w,z = − w ( P ), where w ( P ) stands for the winding number of P . Also note byProposition 6.4, w ( P ) ≤
0. Hence τ = A ( x ) = A ( x ) − w ( P ).The case given in Figure 29 comes combining Figure 26 (3) or (4) and Figure 25(1) or (2). A similar analysis shows τ = A ( x ).In view of the discussion above, it suffices to determine A ( x ). Note x is thefirst intersection of β and α as we traverse upwards along β in the simplified pairing NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , Figure 29.
Combining Figure 26 (3) or (4) and Figure 25 (1) or (2)diagram; denote this oriented arc starting from β ∩ λ and going upward to x by l (see Figure 27), and denote the corresponding arc in the original pairing diagramby ˜ l . Note l · µ = ⌊ w ⌋ according to Proposition 6.4, and the A-buoys on l contributein total to the Alexander grading by ⌈ σ ⌉ − ⌊ w ( P )2 ⌋ = ˜ l · δ w,z . This last equationcan be seen by understanding the effect of L and L (Figure 22) and apply aninductive argument: suppose only a single L occurs throughout; in the begining,as we traverse along β upwards, it intersects µ for ⌈ σ ⌉ times until we reachesthe first intersection point, and each L increases this intersection by 1, and thecorresponding ˜ l · δ w,z = − L has an opposite effect.Let c be the intersection of β and α lying in the center (See Figure 27). We have, A ( c ) − A ( x ) = ( β − ˜ l ) · δ w,z = − w ( P ) − ( ⌈ σ ⌉ − ⌊ w ( P )2 ⌋ ) = − w ( P ) + σ ( P )2 − A ( c ) = 0. Assuming this claim, we have A ( x ) = w ( P )+ σ + 1. It is thenstraightforward to see τ ( P ( K )) = | w ( P ) | + σ ( P )2 + 1 when σ ≥ σ is odd.We now justify our claim on the Alexander grading of c . Figure 30.
The Alexander grading of the center intersection pointis 0.
Proof of the claim A ( c ) = 0 . It is better to have a concrete example in mind, seeFigure 30. Note c corresponds to the center point in the symmetric minimal in-tersection diagram: if we rotate such pairing diagram about c for an angle π , thediagram goes back to itself. We pair the intersection point with its symmetric counterpart. Take a pair of such intersection points y and y ′ , let l cy and l cy ′ denotethe arc on β from c to y and y ′ respectively. Note A ( y ) − A ( c ) = l cy · δ w,z .A ( y ′ ) − A ( c ) = l cy ′ · δ w,z = − l cy ′ · ( − δ w,z )= − l cy ′ · δ z,w = − l cy · δ w,z = A ( c ) − A ( y ) , where in the fourth equality we used the symmetry of the diagram. Therefore theAlexander grading of elements of \ HF K ( P ( T , )) is symmetric about A ( c ), andhence A ( c ) = 0 by the convention of how we grade knot Floer homology groups.This finishes the proof of the claim. (cid:3) We move to consider the case when σ ≤ − τ -invariant. Figure 31.
Possible endings in the upper row when σ ≤ − Figure 32.
Possible endings in the lower row when σ ≤ − NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , First, if the lower row is of type (a) in Figure 32, then the upper row could beof type (2) or type (3) in Figure 31. Figure 33 shows the case when the lower rowis of type (a) and the upper row is of type (2). In this case, 0 ≥ w ( P ) > σ + 2. Ananalysis of the differential as above shows τ = max( A ( x ) , A ( x )). Note we have A ( x ) = 0 (since A ( x ) − A ( c ) = 0) and A ( x ) = w ( P )+ σ + 1 ≤
0. Therefore, τ = 0.Figure 34 shows the case when the lower row is of type (a) and the upper row isof type (3). In this case, w ( P ) ≤ σ + 2. Again τ = max( A ( x ) , A ( x )). While westill have A ( x ) = w ( P )+ σ + 1, A ( x ) = A ( x ) − w ( P ) = − w ( P )+ σ + 1. Therefore, τ = | w ( P ) | + σ + 1. Figure 33.
Lower type (a), upper type (2).
Figure 34.
Lower type (a), upper type (3).Second, if the lower row is of type (b) in Figure 32, then the upper row mustbe of type (1) in Figure 31, and they combine to generate a pairing diagram ofthe form as shown in Figure 35. Note in this case 0 ≤ w ( P ) < − σ −
2. Again, τ = max( A ( x ) , A ( x )). Note we have A ( x ) = 0 and A ( x ) = w ( P )+ σ + 1 ≤ τ = 0.Finally, if the lower row is of type (c) in Figure 32, then the upper row must beof type (1) in Figure 31, and the corresponding pairing diagram is shown in Figure36. Note in this case, w ( P ) ≥ − σ − τ = A ( x ) = τ = w ( P )+ σ + 1. (cid:3) Proof of Corollary 1.9.
First note that if a pattern knot P of winding number w ( P )induces a homomorphism on the smooth knot concordance group, then τ ( P ( K )) = | w ( P ) | τ ( K ). To see this, first note P must be a slice pattern to start with, and Figure 35.
Lower type (b), upper type (1).
Figure 36.
Lower type (c), upper type (1).then a theorem of Roberts in [18] states there is an number ǫ ( P ) ≥ | τ ( P ( K )) − | w ( P ) | τ ( K ) | ≤ ǫ ( P ) for any companion knot K . Suppose τ ( P ( K )) −| w ( P ) | τ ( K ) = 0, then one can choose n sufficiently large so that | τ ( P ( nK )) −| w ( P ) | τ ( nK ) | = n | τ ( P ( K )) − | w ( P ) | τ ( K ) | > ǫ ( P ), which is a contradiction.Now by Theorem 1.7, we may setmax( | w ( P ) | + σ ,
0) = | w ( P ) | min( −| w ( P ) | + σ ,
0) = −| w ( P ) | A simple computation implies both equations hold if and only if | w ( P ) | = 1 and σ = − (cid:3) Brief discussion on immersed curve for general patterns
We give some speculations on how to extend Theorem 1.2 to involve arbitraypattern knots.Without a natural genus-one Heegaard diagram, one has to give a procedure torepresent filtered type D structures by immersed train tracks on T . The difficultyis incurred by the fact that filtered type D structures are often not reduced. Thestrategy given in [2] does not address the unreduced case. In fact, the redundanceof differentials in a type D structure causes two issues. First, not every differetialneeds to be represented by short arcs in the cutted torus [0 , × [0 , NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , labeled by ρ ∅ , one also has to decide on which side of the cutted torus should thecorresponding cap be placed. Figure 37. (a) A filtered type D structure. (b) A filtered exten-sion of (a). Note ρ ∅ is omitted in the label as it is the identity ofthe (extended) torus algebra. Example 7.1.
We illustrate the issues by an example. Consider the filtered type Dstructre given in Figure 37 (a). Here we view the type D structure as a module over
A ⊗ F [ V ], where V is a formal variable used to record the shift of the Alexandergrading in the structure map. (See Page 203 of [10].) If one were to draw thecorresponding train track, then for a disirable pairing theorem to hold, one wouldarrive at Figure 38 (a). (Note when pairing this train track with another, we usedits elliptic involution as in Figure 38 (b) which can be viewed as certain “immersedHeegaard diagram”.) Notice all the edges are drawn using the rules given in Section2.2, but we throw away the edges corresponding to x V ρ −−−→ q and x V ρ −−→ y . Alsoone needs to prevent messing up with the order of p and q , i.e. arriving at a diagramin Figure 39.Such issues can be resolved by introducing a notion called filtered extendability .To spell out, recall the torus algebra A can be extended to a larger algebra ˜ A asshown by the quiver diagram in Figure 40. We use ˜ µ to denote the multiplication,use I to denote the ring of idempotents, and let U denote the central element ρ + ρ + ρ + ρ . Definition 7.2.
A filtered extended type D structure over ˜ A is is a unital left I ⊗ F [ V ]-module N equipped with an I ⊗ F [ V ]-linear map ˜ δ : N → ˜ A ⊗ I N satisfying the compatibility condition(˜ µ ⊗ I ) ◦ ( I ⊗ ˜ δ ) ◦ ˜ δ ( x ) = U V ⊗ x. We point out such condition is satisfied automatically for type D structuresassociated to pattern knots.
Theorem 7.3.
Every filtered type D structure arose from some doubly-pointed bor-dered Heegaard diagram is filtered extendable.
Figure 38. (a) The train track corresponding to the filtered typeD structure in Figure 37 (a). (b) The elliptic involution of (a),which is used when doing Lagrangian intersection pairing; this maybe viewed as an “immersed Heegaard diagram” and used to com-pute the type D structure.
Figure 39. (a) A bad train track representation for the type Dstructure in Figure 37 (a) due to poor position of p and q . (b) Theelliptic involution of (a). Figure 40.
The extended torus algebra
Proof.
Literally the same as Appendix A in [2], with an extra base point taken intoaccount. (cid:3)
NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , Given a filtered extended type D structure, we can associate a train track to itvia the following procedure:(Step 1) Reduce the type D structure until it is filtered reduced , i.e. there is a basisfor N so that over this base, no differential is labeled by ρ ∅ (but could belabeled by V n ρ ∅ for some positive integer n ).(Step 2) Represent the filtered reduced structure by a decorated graph and throwaway all edges corresponding to differentials with nonzero V power.(Step 3) Embed the vertices of the decorated graph into T so that the • vertices lieon α in the interval 0 × [ , ], and the ◦ vertices lie on β in the interval[ , ] × T according to rule as shownin Figure 41. Arrange all the edges to intersect transversely. Figure 41.
Rules for assigning train tracks for a filtered extendedtype D structure.In practice, the immersed train tracks obtained above always form curves, andLagrangian intersection pairing with such curves recovers the box-tensor product.We illustrate the procedure by examples.
Example 7.4.
The filtered extended type D structre shown in Figure 37 (b) is anextension of the type D structure considered in Figure 37 (a). The correspondingtrain track are given in Figure 42: we give two different diagrams correspondingto different ordering of p and q , but curves in the diagrams are obviously regularlyhomotopic to each other. Figure 42.
Train tracks for the filtered extended type D structurein Figure 37
Example 7.5.
Note also that all the filtered type D structures coming from agenus-one doubly-pointed bordered Heegaard diagram can be extended to a filteredextended type D structure, and one can check that if one were to represent suchtype D structures as train tracks by the above algorithm, then one recovers theHeegaard diagram.In general it is easy to see that paring such train tracks with immersed curves ofknot complements would give the \ HF K -group, as the train tracks thus obtainedcorrespond to associated graded objects of the filtered type D structures. It is notclear to the author, though expected, that such train tracks can be represented asimmersed curves. If so, a further question would be if the hat-version filtered knotFloer chain complex can be recovered. The author hope to address these questionsin a future project.
References [1] W. Chen. On the Alexander polynomial and the signature invariant of two-bridge knots. arXiv preprint arXiv:1712.04993 , 2017.[2] J. Hanselman, J. Rasmussen, and L. Watson. Bordered floer homology for manifolds withtorus boundary via immersed curves. arXiv preprint arXiv:1604.03466 , 2016.[3] J. Hanselman, J. Rasmussen, and L. Watson. Heegaard floer homology for manifolds withtorus boundary: properties and examples. arXiv preprint arXiv:1810.10355 , 2018.[4] J. Hanselman and L. Watson. Cabling in terms of immersed curves. arXiv preprintarXiv:1908.04397 , 2019.[5] M. Hedden. Knot Floer homology of Whitehead doubles.
Geom. Topol. , 11:2277–2338, 2007.[6] M. Hedden. On knot Floer homology and cabling. II.
Int. Math. Res. Not. IMRN , 12:2248–2274, 2009.[7] M. Hedden and J. Pinzon-Caicedo. Satellites of infinite rank in the smooth concordancegroup. arXiv preprint arXiv:1809.04186 , 2018.[8] J. Hom. Bordered Heegaard Floer homology and the tau-invariant of cable knots.
J. Topol. ,7(2):287–326, 2014.[9] A. S. Levine. Nonsurjective satellite operators and piecewise-linear concordance.
Forum Math.Sigma , 4:e34, 47, 2016.[10] R. Lipshitz, P. S. Ozsvath, and D. P. Thurston. Bordered Heegaard Floer homology.
Mem.Amer. Math. Soc. , 254(1216):viii+279, 2018.[11] A. N. Miller. Homomorphism obstructions for satellite maps. arXiv preprintarXiv:1910.03461 , 2019.
NOT FLOER HOMOLOGY OF SATELLITE KNOTS WITH (1 , [12] P. J. P. Ording. On knot Floer homology of satellite (1,1) knots . ProQuest LLC, Ann Arbor,MI, 2006. Thesis (Ph.D.)–Columbia University.[13] P. Ozsv´ath and Z. Szab´o. Holomorphic disks and genus bounds.
Geom. Topol. , 8:311–334,2004.[14] P. Ozsv´ath and Z. Szab´o. Holomorphic disks and knot invariants.
Adv. Math. , 186(1):58–116,2004.[15] P. Ozsv´ath and Z. Szab´o. Holomorphic disks and topological invariants for closed three-manifolds.
Ann. of Math. (2) , 159(3):1027–1158, 2004.[16] I. Petkova. Cables of thin knots and bordered Heegaard Floer homology.
Quantum Topol. ,4(4):377–409, 2013.[17] J. A. Rasmussen.
Floer homology and knot complements . ProQuest LLC, Ann Arbor, MI,2003. Thesis (Ph.D.)–Harvard University.[18] L. P. Roberts. Some bounds for the knot Floer τ -invariant of satellite knots. Algebr. Geom.Topol. , 12(1):449–467, 2012.
Max Planck Institute of Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
E-mail address ::