TTHE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCEGROUP
JENNIFER HOM
Abstract.
We define a new smooth concordance homomorphism based on the knot Floer complexand an associated concordance invariant, ε . As an application, we show that an infinite family oftopologically slice knots are independent in the smooth concordance group. Introduction
The set of isotopy classes of knots in S , under the operation of connected sum, forms a monoid.Two knots are concordant if they cobound a smooth, properly embedded cylinder in S × [0 , concordance group , denoted C . If we loosenthe conditions and only require that the cylinder be locally flat, rather than smooth, we obtain the topological concordance group . Understanding the difference between these two groups sheds somelight on the distinction between the smooth and topological categories.Ozsv´ath and Szab´o [OS04], and independently Rasmussen [Ras03], defined an invariant, knotFloer homology, associated to a knot in S . This invariant comes in many different flavors, the mostrobust being CF K ∞ ( K ), a Z -filtered chain complex over the ring F [ U, U − ], where F = Z / Z and U is a formal variable. There is a second filtration induced by − ( U -exponent) allowing us to view CF K ∞ ( K ) as a Z ⊕ Z -filtered chain complex. The filtered chain homotopy type of this complex isan invariant of the knot K . The weaker invariant, (cid:92) CF K ( K ), takes the form of a Z -filtered chaincomplex over F , and is obtained by taking the degree zero part of the associated graded object withrespect to one of the filtrations.Within the complex (cid:92) CF K ( K ) lives a Z -valued concordance invariant, τ ( K ), defined by Ozsv´athand Szab´o in [OS03b]. The total homology of (cid:92) CF K ( K ) has rank one, and τ measures the minimumfiltration level where this homology is supported. The invariant τ gives a surjective homomorphismfrom the smooth concordance group C to the integers: τ : C → Z , which gives a new proof of the Milnor conjecture [OS03b] and is strong enough to obstruct topo-logically slice knots from being smoothly slice (for example, [Liv04]).Often, we would like to be able to show that a collection of n knots is linearly independent, thatis, that they freely generate a subgroup of rank n in C . One way to accomplish this is to define aconcordance homomorphism whose domain has rank at least n , and to show that the image of thiscollection of knots has span equal to n . Thus, the Z -valued concordance homomorphism τ is notsufficient for this type of result.We turn to the more robust invariant CF K ∞ ( K ). In [Hom11a], we defined a {− , , } -valuedconcordance invariant, ε ( K ). The invariant ε is associated to the Z ⊕ Z filtered chain complex CF K ∞ in a manner similar to how τ is associated to the Z -filtered chain complex (cid:92) CF K ; that is,we ask when certain natural maps vanish on homology. We will sometimes write ε ( CF K ∞ ( K )),rather than ε ( K ), to emphasize that ε is an invariant associated to the knot Floer complex of K . a r X i v : . [ m a t h . G T ] N ov JENNIFER HOM
The goal of this paper is to use ε to define a new concordance homomorphism that is strongenough to detect linear independence in C . The main idea is to turn the monoid of chain complexes CF K ∞ ( K ) (under tensor product) into a group, which we will denote F , in much the same waythat the monoid of knots (under connected sum) can be made into the group C by quotienting byslice knots. Definition 1.
Let
CF K ∞ ( K ) ∗ denote the dual of CF K ∞ ( K ) . Define the group F to be F = (cid:0) { CF K ∞ ( K ) | K ⊂ S } , ⊗ (cid:1) / ∼ where CF K ∞ ( K ) ∼ CF K ∞ ( K ) ⇐⇒ ε (cid:0) CF K ∞ ( K ) ⊗ CF K ∞ ( K ) ∗ (cid:1) = 0 . Theorem 2.
The map
C → F , sending a class in C represented by K to the class in F represented by CF K ∞ ( K ) is a grouphomomorphism. This group F has the advantage that it can be studied from an algebraic perspective, much likethe algebraic concordance group defined by Levine [Lev69a, Lev69b] in terms of the Seifert form.However, Levine’s homomorphism factors through the topological concordance group, while oursdoes not.One algebraic feature of F is that it is totally ordered, with an additional well-defined notion ofdomination,“ (cid:28) ”. Moreover, we can use the relation (cid:28) to define a filtration on F that can be usedto show linear independence of certain classes. Given a chain0 < [ CF K ∞ ( K )] (cid:28) [ CF K ∞ ( K )] (cid:28) . . . (cid:28) [ CF K ∞ ( K n )] , it follows that the collection (cid:8) [ CF K ∞ ( K i )] (cid:9) ni =1 is linearly independent in F , and hence (cid:8) [ K i ] (cid:9) ni =1 is independent in C . (It is also possible to use spectral sequences to define a second, independentfiltration on the group F .) One consequence of this filtration is that F contains a subgroupisomorphic to Z ∞ ; see Theorem 3 below. We will use this rich structure on F to better understand C . Let T p,q denote the ( p, q )-torus knot, K p,q the ( p, q )-cable of K (where p denotes the longitudinalwinding and q denotes the meridional winding), and D the (positive, untwisted) Whitehead doubleof the right-handed trefoil. We write T m,n ; p,q to denote the ( p, q )-cable of the ( m, n )-torus knot.Let − K denote the reverse of the mirror image of K , that is, the inverse of K in C . Theorem 3.
The topologically slice knots D p,p +1 − T p,p +1 , p ≥ are independent in the smooth concordance group; that is, they freely generate a subgroup of infiniterank. The first example of an infinite family of smoothly independent, topologically slice knots wasgiven by Endo [End95]. His examples consist of certain pretzel knots. More recently, Hedden andKirk [HK10] showed that an infinite family of (untwisted) Whitehead doubles of certain torus knotsare smoothly independent. The structure of F shows that our examples (when p >
1) are smoothlyindependent from both of these earlier families.
HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 3
Let P ( K ) denote the satellite of K with pattern P ; that is, P is a knot in S × D , which wethen glue into the (zero framed) knot complement S − nbd K to obtain the knot P ( K ) ⊂ S .Recall that the map P ( − ) : C → C given by[ K ] (cid:55)→ [ P ( K )]is well-defined, by “following” the concordance along the satellite.We obtain a similar well-defined map on F : Proposition 4.
The map P ( − ) : F → F given by [ CF K ∞ ( K )] (cid:55)→ [ CF K ∞ (cid:0) P ( K ) (cid:1) ] is well-defined. By composing P with τ , we obtain a new concordance invariant τ P ( K ) = τ (cid:0) P ( K ) (cid:1) , since K being concordant to K implies that P ( K ) is concordant to P ( K ). In the followingtheorem, we relate this to [ CF K ∞ ( K )]. Theorem 5. [ CF K ∞ ( K )] = [ CF K ∞ ( K )] if and only if τ P ( K ) = τ P ( K ) for all patterns P ⊂ S × D . Recall that τ is associated to the weaker, Z -filtered chain complex (cid:92) CF K . The above theorem saysthat knowing information about a weaker invariant, namely τ , of satellites of K tells us informationabout the stronger invariant, CF K ∞ , of the knot itself.Does the map P ( − ) : C → C always take linearly independent collections of knots to linearlyindependent collections of knots? We address this question for cables in the following theorem:
Theorem 6.
For each n ∈ N , there exists a collection of linearly independent knots { K i } ni =1 such that for m ≥ n − n − , { K i , m +1 } ni =1 is a collection of linearly independent knots in C . This result should be compared to the work of Hedden and Kirk [HK10], where they use instantonsto prove that the Whitehead doubles of (2 , n + 1)-torus knots are linearly independent.Central to the definition of F is the concordance invariant ε , which exhibits the following prop-erties: • If K is smoothly slice, then ε ( K ) = 0. • If ε ( K ) = 0, then τ ( K ) = 0. • There exist knots K with τ ( K ) = 0 but ε ( K ) (cid:54) = 0; that is, ε is strictly stronger than τ atobstructing sliceness. • ε ( − K ) = − ε ( K ). • If ε ( K ) = ε ( K (cid:48) ), then ε ( K K (cid:48) ) = ε ( K ). If ε ( K ) = 0, then ε ( K K (cid:48) ) = ε ( K (cid:48) ).These facts are proved in [Hom11a]; we give sketches of their proofs in Section 3. Notice that since ε ( K ) = 0 implies that τ ( K ) = 0, the map τ : C → Z factors through F . JENNIFER HOM
Organization.
We begin by recounting the necessary definitions and properties of the complex
CF K ∞ (Section 2) and the concordance invariant ε (Section 3). With these definitions in place, weproceed to define the group F , describe its various algebraic properties, and give examples (Section4). We study satellites in Section 5. We conclude with the algebraic details in Section 6.We work with coefficients in F = Z / Z throughout. Acknowledgements.
I would like to thank Paul Melvin, Chuck Livingston, Matt Hedden, RumenZarev, Robert Lipshitz, Peter Ozsv´ath, and Dylan Thurston for helpful conversations, and PeterHorn for his comments on an earlier version of this paper.2.
The knot Floer complex
CF K ∞ To a knot K ⊂ S , Ozsv´ath and Szab´o [OS04], and independently Rasmussen [Ras03], associate CF K ∞ ( K ), a Z -filtered chain complex over F [ U, U − ], whose filtered chain homotopy type is aninvariant of K . The complex CF K ∞ can be considered as a Z ⊕ Z -filtered chain complex, withthe second filtration induced by − ( U -exponent). The ordering on Z ⊕ Z is given by ( i, j ) ≤ ( i (cid:48) , j (cid:48) )if i ≤ i (cid:48) and j ≤ j (cid:48) . We assume the reader is familiar with this invariant, and the various relatedflavors, CF K − ( K ) and (cid:92) CF K ( K ); for an expository introduction to these invariants, see [OS06].The knot K is specified by a doubly pointed Heegaard diagram, (Σ , α , β , w, z ), and the generators(over F [ U, U − ]) of CF K ∞ ( K ) are the usual g -tuples of intersection points between the α - and β -circles, where g is the genus of Σ and each α -circle and each β -circle is used exactly once. Thedifferential is defined as ∂x = (cid:88) y ∈ S ( H ) (cid:88) φ ∈ π ( x,y )ind( φ )=1 (cid:99) M ( φ ) U n w ( φ ) · y. This complex is endowed with a homological Z -grading, called the Maslov grading M , as well asa Z -filtration, called the Alexander filtration A . The relative Maslov and Alexander gradings aredefined as M ( x ) − M ( y ) = ind( φ ) − n w ( φ ) and A ( x ) − A ( y ) = n z ( φ ) − n w ( φ ) , for φ ∈ π ( x, y ). The differential, ∂ , decreases the Maslov grading by one, and respects the Alexan-der filtration; that is, M ( ∂x ) = M ( x ) − A ( ∂x ) ≤ A ( x ) . Multiplication by U shifts the Maslov grading and respects the Alexander filtration as follows: M ( U · x ) = M ( x ) − A ( U · x ) = A ( x ) − . It is often convenient to view this complex in the ( i, j )-plane, where the i -axis represents − ( U -exponent) and the j -axis represents the Alexander filtration. The Maslov grading is sup-pressed from this picture. We place a generator x at position (0 , A ( x )); more generally, an elementof the form U i · x will have coordinates ( − i, A ( x ) − i ).A basis { x i } for a filtered chain complex ( C, ∂ ) is called a filtered basis if the set { x i | x i ∈ C S } is a basis for C S for all filtered subcomplexes C S ⊂ C . Given a filtered basis for CF K ∞ , we mayvisualize the differential by placing an arrow from a generator x to a generator y if y appears in ∂x .The differential points non-strictly to the left and down. Often, it will be convenient to consideronly the part of the differential that preserves the Alexander grading, i.e., the horizontal arrows.We will denote this by ∂ horz . Similarly, we will use ∂ vert to denote the part of the differential thatpreserves the filtration by powers of U , i.e., the vertical arrows. HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 5
Given S ⊂ Z ⊕ Z , let C { S } denote the set of elements in the plane whose ( i, j )-coordinates are in S together with the arrows between them. The complex CF K − ( K ) is the subcomplex C { i ≤ } ,that is, the left half-plane. The complex (cid:92) CF K ( K ) is the subquotient complex C { i = 0 } .The integer-valued smooth concordance invariant τ ( K ) is defined in [OS03b] to be τ ( K ) = min { s | ι : C { i = 0 , j ≤ s } → C { i = 0 } induces a non-trivial map on homology } , where ι is the natural inclusion of chain complexes. Alternatively, τ ( K ) may be defined in termsof the U -action on HF K − ( K ), as in [OST08, Appendix A]: τ ( K ) = − max { s | ∃ [ x ] ∈ HF K − ( K, s ) such that ∀ d ≥ , U d [ x ] (cid:54) = 0 } , where HF K − ( K, s ) = H ∗ ( C { i ≤ , j = s } ).The complex CF K ∞ ( K ) satisfies certain symmetry and rank properties [OS04, Section 3]. Thecomplex obtained by interchanging the roles of i and j is filtered chain homotopic to the original.Also, the rank of the homology of any column or row is one; more generally, modulo grading shifts,any column or row is filtered chain homotopic to (cid:92) CF K ( K ).By [OS04, Theorem 7.1], we have the filtered chain homotopy equivalence CF K ∞ ( K K ) (cid:39) CF K ∞ ( K ) ⊗ F [ U,U − ] CF K ∞ ( K ) . Let − K denote the reverse of the mirror image of K . The knot Floer complex is not sensitive tochanges in orientation of the knot, but it is sensitive to changes in the orientation of the ambientmanifold [OS04, Section 3.5]. In particular, CF K ∞ ( − K ) (cid:39) CF K ∞ ( K ) ∗ , where CF K ∞ ( K ) ∗ denotes the dual of CF K ∞ ( K ), i.e., Hom F [ U,U − ] ( CF K ∞ ( K ) , F [ U, U − ]). Todepict the complex CF K ∞ ( K ) ∗ in the ( i, j )-plane, we take the complex CF K ∞ ( K ) and reversethe direction of all of the arrows as well as the direction of both of the filtrations. (In practice,we can accomplish this by reversing the direction of all of the arrows and then turning our headsupside down.)We point out that when we write CF K ∞ ( K ), we are really denoting an equivalence class offiltered chain complexes. We may always choose as our representative the E page of the spectralsequence associated to one of these complexes, that is, the homology of the associated graded objecttogether with the induced differentials. In other words, we may choose our representative to bereduced, in the sense that any differential strictly lowers the filtration (in at least one direction).3. The invariant ε The invariant ε can be defined in terms of the (non-)vanishing of certain cobordism maps, which,using the relation between large surgery and knot Floer homology [OS04, Theorems 4.1 and 4.4],has an algebraic interpretation in terms of the filtered chain complex CF K ∞ ( K ).Let N be a sufficiently large integer. (It turns out that N > g ( K ) will suffice; see [OS08,Theorem 1.1] and [OS04, Theorem 5.1].) We consider the map F s : (cid:100) HF ( S ) → (cid:100) HF ( S − N ( K ) , [ s ]) , induced by the 2-handle cobordism, W − N . As usual, [ s ] denotes the restriction to S − N ( K ) of theSpin c structure s s over W − N with the property that (cid:104) c ( s s ) , [ (cid:98) F ] (cid:105) + N = 2 s, JENNIFER HOM where | s | ≤ N and (cid:98) F denotes the capped off Seifert surface in the four manifold. We also considerthe map G s : (cid:100) HF ( S N ( K ) , [ s ]) → (cid:100) HF ( S ) , induced by the 2-handle cobordism, − W N .The maps F s and G s can be defined algebraically by studying certain natural maps on subquotientcomplexes of CF K ∞ ( K ), as in [OS04]. The map F s is induced by the chain map C { i = 0 } → C { min( i, j − s ) = 0 } consisting of quotienting by C { i = 0 , j < s } , followed by inclusion. Similarly, the map G s is inducedby the chain map C { max ( i, j − s ) = 0 } → C { i = 0 } consisting of quotienting by C { i < , j = s } , followed by inclusion.For ease of notation, we will often write simply τ for τ ( K ) when the meaning is clear fromcontext. Notice that for s > τ , F s is trivial, since quotienting C { i = 0 } by C { i = 0 , j < s } willinduce the trivial map, as the homology of C { i = 0 } is supported in filtration level τ .For s < τ , F s is non-trivial, since any generator of H ∗ ( C { i = 0 } ) will still be in the kernel, butnot the image, of the differential on C { min( i, j − s ) = 0 } .The map F τ may be trivial or non-trivial, depending on whether the class representing a generatorof H ∗ ( C { i = 0 } ) lies in the image of the differential on C { min( i, j − τ ) = 0 } or not.The maps G τ behaves similarly. For s > τ , the map G s is non-trivial, and for s < τ , G s istrivial. The map G τ will be non-trivial if the class representing a generator of H ∗ ( C { i = 0 } lies inthe kernel of the differential on C { max ( i, j − s ) = 0 } , and trivial otherwise.Because C { j = τ } is a chain complex, and so ∂ = 0, it follows that F s and G s cannot bothbe trivial; that is, a class cannot lie in the image but not in the kernel of the differential. (This ismade precise in [Hom11a].) Therefore, there are three possibilities for F τ and G τ : either exactlyone vanishes, or neither vanishes. Definition 3.1.
The invariant ε is defined in terms of F τ and G τ as follows: • ε ( K ) = 1 if and only if F τ is trivial (in which case G τ is necessarily non-trivial). • ε ( K ) = − if and only if G τ is trivial (in which case F τ is necessarily non-trivial). • ε ( K ) = 0 if and only if both F τ and G τ are non-trivial. Let [ x ] be a generator of H ∗ ( C { i = 0 } ), the so-called “vertical” homology. In light of thepreceding discussion, the definition of ε corresponds to viewing [ x ] as a class in the “horizontal”complex C { j = τ } as follows: • ε ( K ) = 1 if and only if [ x ] is in the image of horizontal differential. • ε ( K ) = − x ] is not in the kernel of the horizontal differential. • ε ( K ) = 0 if and only if [ x ] is in the kernel but not the image of the horizontal differential.Notice that ε is an invariant of the filtered chain homotopy type of CF K ∞ ; at times, to emphasizethis point, we will write ε ( CF K ∞ ( K )) rather than simply ε ( K ).This idea of associating numerical invariants to filtered chain complexes is common; for example,to any Z -filtered chain complex whose total homology has rank one, we can define an integer-valued invariant that measures the minimum filtration level at which this homology is supported,e.g., τ ( K ), which is an invariant of the Z -filtered chain homotopy type of (cid:92) CF K ( K ).Similarly, to any Z ⊕ Z -filtered chain complex whose “vertical” homology has rank one, wecan define a {− , , } -valued invariant that measures how this class appears in the “horizontal”complex, i.e., in the image of the horizontal differential, in the kernel but not the image, or not in the HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 7 kernel, respectively. In particular, when ε ( K ) = 0, then CF K ∞ ( K ) is filtered chain homotopic toa complex with a distinguished generator that is non-trivial in both the vertical and the horizontalhomology. Proposition 3.2 ([Hom11a]) . The following are properties of ε ( K ) : (1) If K is smoothly slice, then ε ( K ) = 0 . (2) If ε ( K ) = 0 , then τ ( K ) = 0 . (3) ε ( − K ) = − ε ( K ) . (4) (a) If ε ( K ) = ε ( K (cid:48) ) , then ε ( K K (cid:48) ) = ε ( K ) = ε ( K (cid:48) ) . (b) If ε ( K ) = 0 , then ε ( K K (cid:48) ) = ε ( K (cid:48) ) . For completeness, we sketch the proof below.
Sketch of proof.
To prove (1), we consider the d -invariants of large surgery along K . If K is slice,then the surgery correction terms defined in [OS03a] vanish, i.e., agree with the surgery correctionterms of the unknot, and the maps (cid:100) HF ( S N ( K ) , [0]) → (cid:100) HF ( S ) and (cid:100) HF ( S ) → (cid:100) HF ( S − N ( K ) , [0])are non-trivial. Indeed, the surgery corrections terms can be defined in terms of the maps HF + ( S ) → HF + ( S − N ( K ) , [ s ])and we have the commutative diagram (cid:100) HF ( S ) F s −−−−→ (cid:100) HF ( S − N ( K ) , [ s ]) ι (cid:121) ι s (cid:121) HF + ( S ) F + s −−−−→ HF + ( S − N ( K ) , [ s ]) . Let N (cid:29)
0. If the surgery corrections terms vanish (that is, agree with those of the unknot),then F + τ is an injection [Ras04, Section 2.2], and so the composition ι ◦ F + τ is non-trivial. Bycommutativity of the diagram, it follows that F τ must be non-trivial. A similar diagram in thecase of large positive surgery shows that G τ must be non-trivial as well. Hence ε ( K ) = 0.The proof of (2) follows from the fact that if ε ( K ) = 0, then there is a class x in CF K ∞ ( K )which generates both H ∗ ( C { i = 0 } ) and H ∗ ( C { j = 0 } ). In the former complex, x has Alexandergrading A ( x ), and in the latter, viewed as a Z -filtered complex, x has filtration level − A ( x ). Hence τ ( K ) = − τ ( K ) = 0.The proof of (3) follows from the symmetry properties of the knot Floer complex [OS04, Section3.5]; in particular, we have the filtered chain homotopy equivalence CF K ∞ ( − K ) (cid:39) CF K ∞ ( K ) ∗ .To prove the first part of (4): if [ x ] and [ x (cid:48) ] are generators of H ∗ ( (cid:92) CF K ( K )) and H ∗ ( (cid:92) CF K ( K (cid:48) )),respectively, then [ x ⊗ x (cid:48) ] is a generator of H ∗ ( (cid:92) CF K ( K K (cid:48) )). (Here, we are identifying (cid:92) CF K with C { i = 0 } .) Suppose ε ( K ) = ε ( K (cid:48) ) = 1. Then both [ x ] and [ x (cid:48) ] are both in the image of thehorizontal differential, and hence [ x ⊗ x (cid:48) ] is also. The other cases follow similarly. (cid:3) Notice that Proposition 3.2 implies that ε is a concordance invariant. If K and K (cid:48) are concordant,then ε ( K − K (cid:48) ) = 0, in which case ε ( K ) = − ε ( − K (cid:48) ) by (4), or ε ( K ) = ε ( K (cid:48) ).Note that we have the following subgroup of C : { [ K ] | ε ( K ) = 0 } ⊂ C . This observation will useful in the next section.
JENNIFER HOM The group F In this section, we define the group F as well as some of its algebraic structure. We will giveexamples of knots that demonstrate the richness of this structure. In particular, we give an infinitefamily of topologically slice knots that are linearly independent in F , and hence also in the smoothconcordance group C , as needed for the proof of Theorem 3.4.1. Definition of the group F . We define the group F as F = (cid:0) { CF K ∞ ( K ) | K ⊂ S } , ⊗ (cid:1) / ∼ , where CF K ∞ ( K ) ∼ CF K ∞ ( K ) ⇐⇒ ε (cid:0) CF K ∞ ( K ) ⊗ CF K ∞ ( K ) ∗ (cid:1) = 0 ,CF K ∞ ( K ) ∗ denotes the dual of CF K ∞ ( K ), and the tensor product is over F [ U, U − ]. We havethe well-defined group homomorphism C → F , given by [ K ] (cid:55)→ [ CF K ∞ ( K )] . Well-definedness follows from the following facts (the first two from [OS04, Section 3.5] and thelast from Proposition 3.2): • CF K ∞ ( − K ) (cid:39) CF K ∞ ( K ) ∗ . • CF K ∞ ( K K ) (cid:39) CF K ∞ ( K ) ⊗ CF K ∞ ( K ). • If K is smoothly slice, then ε ( CF K ∞ ( K )) = 0.Notice that F is isomorphic to the quotient F ∼ = C / { [ K ] | ε ( K ) = 0 } . For ease of notation, from now on, we will write (cid:74) K (cid:75) to denote [ CF K ∞ ( K )], and, when convenient, we will write (cid:74) K (cid:75) + (cid:74) K (cid:75) to denote the operation on the group, which can be thought of as either [ CF K ∞ ( K ) ⊗ CF K ∞ ( K )]or [ CF K ∞ ( K K )]. Note that − (cid:74) K (cid:75) = (cid:74) − K (cid:75) . We denote the identity of F , (cid:74) unknot (cid:75) , by 0.The group F has a rich algebraic structure: it has a total ordering, and a “ (cid:28) ” relation thatsatisfies the expected properties and induces a filtration on the group. This algebraic structure on F will in turn be useful in understanding the structure of the smooth concordance group C . Proposition 4.1.
The group F is totally ordered, with the ordering given by (cid:74) K (cid:75) > (cid:74) K (cid:75) ⇐⇒ ε ( K − K ) = 1 . Proof.
We may think of ε ( K ) as the “sign” of (cid:74) K (cid:75) , and then the order relation between any twoclasses is determined by the sign of their difference.This relation is clearly transitive, since given (cid:74) K (cid:75) > (cid:74) K (cid:75) and (cid:74) K (cid:75) > (cid:74) K (cid:75) , it follows that (cid:74) K (cid:75) > (cid:74) K (cid:75) . HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 9
Indeed, ε ( K − K ) = ε ( K − K K − K )= 1 , by (4) of Proposition 3.2 since ε ( K − K ) = 1 and ε ( K − K ) = 1.This relation is also translation invariant. Given (cid:74) K (cid:75) > (cid:74) K (cid:75) , it follows that (cid:74) K (cid:75) + (cid:74) K (cid:75) > (cid:74) K (cid:75) + (cid:74) K (cid:75) , since ε ( K K − K − K ) = ε ( K − K )= 1 . (cid:3) Totally ordered groups give rise to many natural algebraic constructions, which we will utilizebelow. For example, we have a notion of absolute value; that is, given an element (cid:74) K (cid:75) , either (cid:74) K (cid:75) or − (cid:74) K (cid:75) is greater than the identity, so we define the absolute value as (cid:12)(cid:12) (cid:74) K (cid:75) (cid:12)(cid:12) = (cid:26) (cid:74) K (cid:75) if ε ( K ) ≥ − (cid:74) K (cid:75) otherwise . A natural question to ask is: Do there exist knots K and K with ε ( K ) = ε ( K ) = 1 (i.e., theyare both “positive” with respect to the ordering), and (cid:74) K ] > n (cid:74) K (cid:75) for all n ∈ N ?The answer, it turns out, is yes, motivating the following definition: Definition 4.2.
The class (cid:74) K (cid:75) dominates (cid:74) K (cid:75) , denoted (cid:74) K (cid:75) (cid:29) (cid:74) K (cid:75) , if (cid:74) K ] > n (cid:74) K (cid:75) > n ∈ N . Transitivity of (cid:29) follows exactly as for the total ordering. We have the following lemma, showingthat the (cid:29) relation satisfies a property we would expect of a “much bigger” relation:
Lemma 4.3. If (cid:74) K (cid:75) (cid:29) (cid:74) K (cid:75) and (cid:74) K (cid:75) (cid:29) (cid:74) K (cid:75) then (cid:74) K (cid:75) (cid:29) (cid:74) K (cid:75) + (cid:74) K (cid:75) . Proof.
To see that this is true, we proceed by contradiction. Assume there exists n ∈ N such that (cid:74) K (cid:75) ≤ n (cid:0) (cid:74) K (cid:75) + (cid:74) K (cid:75) (cid:1) . Then 2 (cid:74) K (cid:75) ≤ n (cid:0) (cid:74) K (cid:75) + (cid:74) K (cid:75) (cid:1) , i.e., (cid:74) K (cid:75) − n (cid:74) K (cid:75) + (cid:74) K (cid:75) − n (cid:74) K (cid:75) ≤ . But (cid:74) K (cid:75) − m (cid:74) K (cid:75) > (cid:74) K (cid:75) − m (cid:74) K (cid:75) > m ∈ N , giving us the desired contradiction. (cid:3) Remark 4.4.
These ideas could alternatively be phrased in terms of Archimedean equivalenceclasses. Recall that two elements a and b of a totally ordered group are Archimedean equivalent ifthere exist natural numbers M and N such that M · | a | > | b | and N · | b | > | a | . Then we say that a (cid:29) b if a > b >
0, and a and b are not Archimedean equivalent. Note that the set of Archimedeanequivalence classes is naturally totally ordered, and this ordering corresponds to the (cid:29) relation. Definition 4.5.
Let F K denote the collection of elements F K = { (cid:74) J (cid:75) (cid:12)(cid:12) | (cid:74) J (cid:75) | (cid:28) | (cid:74) K (cid:75) |} . Proposition 4.6. F K is a subgroup of F .Proof. If (cid:74) J (cid:75) is in F K , then − (cid:74) J (cid:75) clearly is as well. Given (cid:74) J (cid:75) and (cid:74) J (cid:75) in F K , is follows immediatelythat (cid:74) J (cid:75) + (cid:74) J (cid:75) is also in F K , by Lemma 4.3. (cid:3) Notice that given a sequence of knots K , K , . . . , K n satisfying (cid:74) K (cid:75) (cid:29) (cid:74) K (cid:75) (cid:29) . . . (cid:29) (cid:74) K n (cid:75) , we obtain a filtration F K ⊃ F K ⊃ . . . ⊃ F K n . Lemma 4.7. If (cid:74) K (cid:75) (cid:29) (cid:74) K (cid:75) (cid:29) . . . (cid:29) (cid:74) K n (cid:75) > , then the knots K , K , . . . , K n are linearly independent in F and hence in C ; that is, they generate a subgroup of rank n in both F and C .Proof. By Lemma 4.3, for any positive integer m , m (cid:74) K (cid:75) dominates any linear combination of (cid:74) K (cid:75) , . . . , (cid:74) K n (cid:75) , and thus cannot be expressed as a linear combination of these classes. Similarly, m (cid:74) K i (cid:75) dominates any linear combination of (cid:74) K i +1 (cid:75) , . . . , (cid:74) K n (cid:75) , for i < n . (cid:3) Examples.
We now give examples of families of knots that can be shown to independent in C . Proposition 4.8.
Let < p < q . Then we have the following relations in the group F : (1) (cid:74) T p,p +1 (cid:75) (cid:28) (cid:74) T q,q +1 (cid:75) (2) (cid:74) D p,p +1 (cid:75) (cid:28) (cid:74) D q,q +1 (cid:75) (3) (cid:74) T p,p +1 (cid:75) (cid:28) (cid:74) D p,p +1 (cid:75) (4) (cid:74) T p,p +1;2 , m +1 (cid:75) (cid:28) (cid:74) T q,q +1;2 , m +1 (cid:75) , m ≥ q − q − . We will prove this proposition at the end of Section 6.
Remark 4.9.
A straightforward consequence of (2) and (3) of the preceding proposition is therelation (cid:74) D p,p +1 − T p,p +1 (cid:75) (cid:28) (cid:74) D q,q +1 − T q,q +1 (cid:75) . We are now ready to prove Theorem 3; that is, we will show that the knots D p,p +1 − T p,p +1 , p ≥ HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 11
Proof of Theorem 3.
Recall that D is the (positive, untwisted) Whitehead double of the right-handed trefoil. The Alexander polynomial of D is equal to one, and so by Freedman [FQ90],it follows that D is topologically slice. Hence, the ( p, p + 1)-cable of D , D p,p +1 , is topologicallyconcordant to the ( p, p + 1)-cable of the unknot, i.e., the torus knot T p,p +1 . Thus, D p,p +1 − T p,p +1 is topologically slice.It follows from Lemma 4.7 and Remark 4.9 that the knots D p,p +1 − T p,p +1 , p ≥ F , and hence also in C . (cid:3) Proof of Theorem 6.
We need to find a collection of linearly independent knots { K i } ni =1 such thatthe collection { K i , m +1 } ni =1 is also linearly independent for sufficiently large m .Let K i = T i,i +1 , and consider the (2 , m + 1)-cable of K i , where m ≥ n − n −
1. By Lemma 4.7and Proposition 4.8, it follows that the collection { K i } ni =1 is linearly independent in F , hence also in C . Again, by Lemma 4.7 and Proposition 4.8, thecollection { K i , m +1 } ni =1 , is also linearly independent in F and thus in C . (cid:3) Satellites and F Recall that P ( K ) denotes the satellite of K with pattern P ; that is, P is a knot in S × D ,which we then glue into the (zero framed) knot complement S − nbd K . The map P ( − ) : C → C given by [ K ] (cid:55)→ [ P ( K )]is well-defined, by “following” the concordance along the satellite. We will show that an analogousresult holds for the group F . Proposition 5.1.
The map P ( − ) : F → F given by (cid:74) K (cid:75) (cid:55)→ (cid:74) P ( K ) (cid:75) is well-defined. The following theorem from [Hom11a] gives a formula for τ ( K p,q ) in terms of τ ( K ), ε ( K ), p , and q : Theorem 5.2 ([Hom11a]) . Let K ⊂ S , and let p , q be relatively prime integers with p > . Thenthe behavior of τ ( K p,q ) is completely determined by p , q , τ ( K ) , and ε ( K ) . More precisely: (1) If ε ( K ) = 1 , then τ ( K p,q ) = pτ ( K ) + ( p − q − . (2) If ε ( K ) = − , then τ ( K p,q ) = pτ ( K ) + ( p − q +1)2 . (3) If ε ( K ) = 0 , then τ ( K ) = 0 and τ ( K p,q ) = τ ( T p,q ) = (cid:40) ( p − q +1)2 if q < ( p − q − if q > . We see that knowing τ ( K , ) and τ ( K , − ) is sufficient to determine ε ( K ). More precisely, • If τ ( K , ) is odd, then ε ( K ) = − • If τ ( K , − ) is odd, then ε ( K ) = 1. • If τ ( K , ) = τ ( K , − ) = 0, then ε ( K ) = 0. The proof of Proposition 5.1 will rely on this observation.The proof will also rely on facts from bordered Heegaard Floer homology, as defined by Lipshitz,Ozsv´ath and Thurston [LOT08]. We will need only a special case of the formal properties of theseinvariants, which we recount here.To a framed knot complement Y K , we associate a left differential graded module (cid:92) CF D ( Y K ),whose homotopy equivalence class is an invariant of the framed knot complement [LOT08, Theo-rem 1.1]. Furthermore, the homotopy equivalence class is completely determined by the complex CF K ∞ ( K ) and the framing n [LOT08, Theorem 11.27 and A.11]. For our purposes here, it will besufficient to let Y K be the zero framed knot complement. In [Hom11a], it is shown that if ε ( K ) = 0,then (cid:92) CF D ( Y J K ) (cid:39) (cid:92) CF D ( Y J ) ⊕ A, for some left differential graded module A .To a knot P in S × D , we associate a right A ∞ -module CF A − ( S × D , P ). Let gCF K − ( K )denote the associated graded complex of CF K − ( K ), i.e., ⊕ s C { i ≤ , j = s } . Notice that HF K − ( K ) ∼ = H ∗ ( gCF K − ( K )). Then the pairing theorem for bordered Heegaard Floer homology[LOT08, Theorem 11.21] states that we have the following graded chain homotopy equivalence: gCF K − ( S , P ( K )) (cid:39) CF A − ( S × D , P ) (cid:101) ⊗ (cid:92) CF D ( Y K ) , where we choose the zero framing for the knot complement Y K , and where (cid:101) ⊗ denotes the A ∞ -tensorproduct, a generalization of the derived tensor product. In particular, (cid:101) ⊗ respects summands. Proof of Proposition 5.1.
Assume ε ( K − J ) = 0. We would like to show that ε ( P ( K ) − P ( J )) = 0 . Utilizing the observation above, it is sufficient to show that τ (cid:0) ( P ( K ) − P ( J )) , ± (cid:1) = 0 . K K
Figure 1.
The knot (cid:0) P ( K ) − P ( K ) (cid:1) , , in the case where P is the pattern forthe Whitehead double.Let U denote the unknot. There exists an embedding Q of (cid:0) P ( U ) − P ( J ) (cid:1) , ± into S × D such that Q ( K ) = (cid:0) P ( K ) − P ( J ) (cid:1) , ± . HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 13 K Figure 2.
The knot (cid:0) P ( U ) − P ( K ) (cid:1) , embedded in S × D as the pattern knot Q , where again, P is the pattern for the Whitehead double.See Figure 2. We consider the bordered invariant CF A − ( S × D , Q )associated to ( S × D , Q ). Notice that Q ( J ) = (cid:0) P ( J ) − P ( J ) (cid:1) , ± is slice and so τ (cid:0) ( P ( J ) − P ( J )) , ± (cid:1) = 0.The knot K is concordant to the knot K (cid:48) = J K − J . Since ε ( K − J ) = 0, we have thefollowing chain homotopy equivalence: (cid:92) CF D ( Y K (cid:48) ) (cid:39) (cid:92) CF D ( Y J ) ⊕ A, for some A .The knot Q ( K ) is concordant to Q ( K (cid:48) ), since K is concordant to K (cid:48) . The invariant τ ( Q ( K (cid:48) )) isdetermined by gCF K − ( Q ( K (cid:48) )) (cid:39) CF A − ( S × D , Q ) (cid:101) ⊗ (cid:92) CF D ( Y K (cid:48) ) (cid:39) CF A − ( S × D , Q ) (cid:101) ⊗ (cid:0) (cid:92) CF D ( Y J ) ⊕ A (cid:1) (cid:39) gCF K − (cid:0) Q ( J ) (cid:1) ⊕ B where B is the complex CF A − ( S × D , Q ) (cid:101) ⊗ A . Notice that H ∗ ( B ) is U -torsion, since the ranksof HF K − ( Q ( K (cid:48) )) and HF K − ( Q ( J )) as F [ U ]-modules are both one. Thus, τ ( Q ( K )) = τ ( Q ( K (cid:48) )) = τ ( Q ( J )) = 0 , since Q ( J ) is slice. Recalling that Q ( K ) = (cid:0) P ( K ) − P ( J ) (cid:1) , ± , we have that τ (cid:0) ( P ( K ) − P ( J )) , ± (cid:1) = 0 , implying that ε ( P ( K ) − P ( J )) = 0 , as desired. (cid:3) We now prove Theorem 5, which we restate here:
Theorem 5.3. (cid:74) K (cid:75) = (cid:74) J (cid:75) if and only if τ P ( K ) = τ P ( J ) for all patterns P ⊂ S × D .Proof. The forward direction is true by Proposition 5.1 and the fact that the map τ : C → Z factorsthrough F . We must now show that if (cid:74) K (cid:75) (cid:54) = (cid:74) J (cid:75) , then there exists some pattern P such that τ ( P ( K )) (cid:54) = τ ( P ( J )).Without loss of generality, we may assume that ε ( K − J ) = −
1. Let P ( K ) = ( K − J ) , . Then Theorem 5.2 tells us that τ ( P ( K )) = 2( τ ( K ) − τ ( J )) + 1 and τ ( P ( J )) = 0 , as desired. (cid:3) Calculations and a refinement of ε An element of F is an equivalence class of filtered chain complexes. The goal of this section isto define more tractable invariants associated to such a class, compute these invariants for a fewfamilies of knots, and show that these invariants are related to the algebraic structure, namely the (cid:29) relation, on F .To this end, we will define a refinement of ε . Recall that ε is defined in terms of whether or notcertain maps on subquotient complexes of CF K ∞ vanish on homology. Our refinement of ε willbe defined in a similar manner.The invariant ε ( K ) is equal to one when the class generating the “vertical” homology of CF K ∞ ( K )lies in the image of the horizontal differential. We would like a well-defined way to measure the“length” of the differential that hits that class, that is, how much it decreases the horizontal filtra-tion. We will do this by examining certain natural maps on subquotients of CF K ∞ .The definition of ε involved examining the map F τ induced by C { i = 0 } → C { min( i, j − τ ) = 0 } . In particular, if F τ is trivial, then ε ( K ) = 1. Consider now the map H s induced on homology by C { i = 0 } → C { min( i, j − τ ) = 0 , i ≤ s } , for some non-negative integer s . Notice that H is non-trivial, and for sufficiently large s , H s agreeswith F τ .Suppose that ε ( K ) = 1; that is, F τ is trivial. Then define a ( K ) to be a ( K ) = min { s | H s is trivial } . The idea is that when ε ( K ) = 1, the class generating the vertical homology lies in the image of thehorizontal differential, and a is measuring the “length” of the horizontal differential hitting thatclass.Now consider the map H a ,s induced on homology by C { i = 0 } → C (cid:8) { min( i, j − τ ) = 0 , i ≤ a } ∪ { i = a , τ − s ≤ j < τ } (cid:9) , for some non-negative integer s . Clearly, H a , is trivial. Define a ( K ) = min { s | H a ,s is non-trivial } . HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 15
Notice that a ( K ) may be undefined; that is, the map H a ,s may be trivial for all s . Effectively, a is measuring the “length” of a certain vertical differential, if it exists. Lemma 6.1.
The invariants a and a are invariants of the class (cid:74) K (cid:75) .Proof. Suppose (cid:74) J (cid:75) = (cid:74) K (cid:75) . Then (cid:74) J (cid:75) = (cid:74) K (cid:75) = (cid:74) K − J J (cid:75) . Since ε ( K − J ) = 0, it follows from [Hom11a, Lemma 3.3] that there exists a basis for CF K ∞ ( K − J ) with a distinguished element, say x , with no incoming or outgoing horizontal or vertical ar-rows. Similarly, there is a basis for CF K ∞ ( J − J ) with a distinguished element y . Then wemay compute a ( K − J J ) and a ( K − J J ) by considering either { x } ⊗ CF K ∞ ( J ) or CF K ∞ ( K ) ⊗ { y } , the former giving us a ( J ) and a ( J ), and the latter giving us a ( K ) and a ( K ). (cid:3) τ a (a) τ a a (b) x x x (c) Figure 3.
Left, the complex A in the ( i, j )-plane. Center, the complex B . Right,part of the basis in Lemma 6.2. Lemma 6.2.
Let a = a ( K ) . Then there exists a basis { x i } over F [ U, U − ] for CF K ∞ with basiselements x and x with the property that (1) There is a horizontal arrow of length a from x to x . (2) There are no other horizontal or vertical arrows to or from x . (3) There are no other horizontal arrows to or from x .If we also have that a = a ( K ) is well-defined, then there exists a basis { x i } with basis elements x , x , and x with the following properties, in addition to the ones above: (4) There is a vertical arrow of length a from x to x . (5) There are no other vertical arrows to or from x or x . Proof.
We will give the proof for the case where a is well-defined. The proof in the case where a is not well-defined is a straightforward simplification of this proof.For ease of notation, let A = C { min( i, j − τ ) = 0 , i ≤ a } B = C (cid:8) { min( i, j − τ ) = 0 , i ≤ a } ∪ { i = a , τ − a ≤ j < τ } (cid:9) , so that H a and H a ,a , respectively, are the maps on homology induced by C { i = 0 } → AC { i = 0 } → B. See Figure 3. Since H a is trivial, it follows that there is a generator, say x , of H ∗ ( C { i = 0 } ) inposition (0 , τ ) that is in the image of the differential on A , but not in the image of the differential on B . Since H a ,a is non-trivial, there exists a class x supported in position ( a , τ ) whose boundary,in A , is x , and whose boundary, in B , is a class, say x + x , where x is supported in position( a , τ − a ). Moreover, we may replace x with ∂ horz x , since a priori, ∂ horz x might include elementswith negative i -coordinate. Similarly, we may replace x with ∂ vert x .We now complete { x , x , x } to a basis { x i } for CF K ∞ ( K ), and conditions (1) and (4) aboveare satisfied. To satisfy the remaining three conditions, we will use a change of basis in order toremove the unwanted arrows.There are no vertical arrows leaving x , since it is in the kernel of the vertical differential. Since x is not in the image of the vertical differential, if there is an incoming vertical arrow to x from,say, y , then there is also a vertical arrow from y to, say, z . Changing basis to replace z with z + x will remove the vertical arrow to x . All of the incoming vertical arrows to x may be removed inthis manner, and filtration considerations ensure that we have not changed x or x .Since x is in the image of ∂ horz , it follows immediately that there are no horizontal arrowsleaving x , by the fact that ∂ horz ◦ ∂ horz = 0. We must now remove any horizontal arrows entering x . Suppose there is an arrow of length (cid:96) from y to x . If (cid:96) < a , we may remove the arrow as inthe preceding paragraph. If (cid:96) ≥ a , then we replace y with y + x . In this manner, we can removeall of other horizontal arrows into x .There are now no horizontal arrows entering x , because ∂ horz x = x , ∂ horz ◦ ∂ horz = 0, andthere are no other horizontal arrows to x .We may remove unwanted vertical arrows involving x and x in the same manner that weremoved unwanted horizontal arrows involving x and x . (cid:3) Note that if we have such a basis { x i } for CF K ∞ ( K ), then we have a basis { x ∗ i } for CF K ∞ ( K ) ∗ satisfying the following: • There is a horizontal arrow of length a ( K ) from x ∗ to x ∗ . • There is a vertical arrow of length a ( K ) from x ∗ to x ∗ . • There are no other horizontal or vertical arrows to or from x ∗ . • There are no other horizontal or vertical arrows to or from x ∗ . • There are no other vertical arrows to or from x ∗ .If x k has filtration level ( i, j ), then x ∗ k has filtration level ( − i, − j ). We will use these types of basesto prove the following lemmas: Lemma 6.3. If a ( J ) > a ( K ) , then (cid:74) K (cid:75) (cid:29) (cid:74) J (cid:75) . HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 17
Proof.
We proceed using induction. We will show that ε ( K − J ) = 1 and that a ( K − J ) = a ( K )from which we can conclude that ε ( K − nJ ) = 1for all n ∈ N .Let { x i } be a basis for CF K ∞ ( K ) found using the first part of Lemma 6.2. Similarly, let { y i } be such a basis for CF K ∞ ( J ), and hence { y ∗ i } is a basis for CF K ∞ ( − J ). We consider the knot K − J and its knot Floer complex. Notice that x y ∗ generates H ∗ ( C { i = 0 } ), the “vertical”homology of CF K ∞ ( K − J ). Let τ = τ ( K − J ).Consider the subquotient complex A = C { min( i, j − τ ) = 0 } . There is a direct summand of A consisting of generators x y ∗ and x y ∗ , with a horizontal arrow oflength a ( K ) from the latter to the former. Hence, ε ( K − J ) = 1 and a ( K − J ) = a ( K ), asdesired. (cid:3) Lemma 6.4. If a ( J ) = a ( K ) and a ( J ) > a ( K ) , then (cid:74) J (cid:75) (cid:29) (cid:74) K (cid:75) . Proof.
We again proceed using induction. We will show that ε ( J − K ) = 1 and that a ( J − K ) = a ( J ) a ( J − K ) = a ( J ) , from which we can conclude that ε ( J − nK ) = 1for all n ∈ N .Let { x i } be a basis for CF K ∞ ( K ) found using Lemma 6.2. Similarly, let { y i } be such a basisfor CF K ∞ ( J ). We consider the knot J − K and its knot Floer complex. For ease of notation,let τ = τ ( J − K ).Let A = C { min( i, j − τ ) = 0 , i ≤ a ( J ) } B = C (cid:8) { min( i, j − τ ) = 0 , i ≤ a ( J ) } ∪ { i = a ( J ) , τ − a ( J ) ≤ j < τ } (cid:9) , We claim that the element x ∗ y + x ∗ y generates H ∗ ( C { i = 0 } ), is zero in H ∗ ( A ), and is non-zeroin H ∗ ( B ). Indeed, there is a direct summand of B with the following generators in the following( i, j )-positions: x ∗ y , x ∗ y (cid:0) , τ ( J − K ) (cid:1) x ∗ y (cid:0) a ( J ) , τ ( J − K ) (cid:1) x ∗ y (cid:0) a ( J ) , τ ( J − K ) − a ( J ) (cid:1) x ∗ y (cid:0) , τ ( J − K ) + a ( K ) (cid:1) , and the following differentials: ∂ ( x ∗ y ) = x ∗ y + x ∗ y + x ∗ y ∂ ( x ∗ y ) = x ∗ y . See Figure 4(d). From this observation, the claim readily follows; that is, ε ( J − K ) = 1 a ( J − K ) = a ( J ) a ( J − K ) = a ( J ) , as desired. (cid:3) x x x (a) x ∗ x ∗ x ∗ (b) y y y (c) x ∗ y x ∗ y x ∗ y x ∗ y x ∗ y (d) Figure 4.
Far left, a portion of the basis { x i } for CF K ∞ ( K ), followed by a portionof the basis { x ∗ i } for CF K ∞ ( K ) ∗ . Next, a portion of the basis { y i } for CF K ∞ ( J ).Far right, a direct summand of the subquotient complex B .Recall that an L -space is a rational homology sphere Y for whichrk (cid:100) HF ( Y ) = | H ( Y, Z ) | . We call a knot K ⊂ S an L -space knot if there exists n ∈ N such that n surgery on K yields an L -space. In [OS05, Theorem 1.2], Ozsv´ath and Szab´o prove that if K is an L -space knot, thenthe complex CF K ∞ ( K ) has a particularly simple form that can be deduced form the Alexanderpolynomial of K , ∆ K ( t ). (Note that the results in [OS05] are stated in terms of (cid:92) HF K ( K ), but byconsidering gradings, they are actually sufficient to determine the full CF K ∞ ( K ) complex.)One consequence is that if K is an L -space knot, then the Alexander polynomial of K has theform ∆ K ( t ) = k (cid:88) i =0 ( − i t n i , for some decreasing sequence of non-negative integers n > n > . . . > n k with the symmetrycondition n i + n k − i = 2 g ( K ) , where we have normalized the Alexander polynomial to have a constant term and no negativeexponents. Note that k is always even since there are always an odd number of terms in theAlexander polynomial. HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 19
Lemma 6.5.
Let K be an L -space knot with Alexander polynomial ∆ K ( t ) = k (cid:88) i =0 ( − i t n i , for some decreasing sequence of integers n > n > . . . > n k . Then a ( K ) = n − n a ( K ) = n − n . Proof.
Theorem 1.2 of [OS05] tells us that for K an L -space knot, (cid:92) HF K ( K ) is completely deter-mined by ∆ K ( t ). Moreover, up to filtered chain homotopy equivalence, CF K ∞ ( K ) is generated asa F [ U, U − ]-module by (cid:92) HF K ( K ), where (cid:92) HF K is the homology of the associated graded object of (cid:92)
CF K ( K ) (cid:39) C { i = 0 } . By considering the gradings on the complex CF K ∞ ( K ), and the fact thatthe differential decreases the Maslov grading by one, the lemma follows. (cid:3) Remark 6.6.
More generally, it can be deduced from [OS05, Theorem 1.2] that there is a basis { x , . . . x k } for CF K ∞ ( K ) such that ∂x i = x i − + x i +1 for i odd ∂x i = 0 otherwise , where the arrow from x i to x i − is horizontal of length n i − n i − , and the arrow from x i to x i +1 isvertical of length n i +1 − n i . The complex looks like a “staircase”, where the differences of the n i give the heights and widths of the steps. See Figure 5.Recall that positive torus knots are L -space knots since ( pq ± T p,q , p, q >
1, results in a lens space.
Lemma 6.7.
For p ≥ , the Alexander polynomial of the torus knot T p,p +1 is ∆ T p,p +1 ( t ) = k (cid:88) i =0 ( − i t n i , for a decreasing sequence of integers n > n > . . . > n k with n = p − pn = p − p − n = p − pn = p − p − . In particular, a ( T p,p +1 ) = 1 a ( T p,p +1 ) = p − . Proof.
Recall that ∆ T p,q ( t ) = ( t pq − t − t p − t q − . Following the proof of Proposition 6.1 in [HLR10], we see that( t p ( p +1) − t − t p − t p +1 −
1) = p − (cid:88) i =0 t pi − t p − (cid:88) i =0 t ( p +1) i . x x x x x (a) x x x x x x x (b) Figure 5.
Left, the basis from Remark 6.6 for
CF K ∞ of the torus knot T , withAlexander polynomial ∆ T , ( t ) = t − t + t − t + 1. Right, the basis for CF K ∞ ofthe torus knot T , with Alexander polynomial ∆ T , ( t ) = t − t + t − t + t − t +1.The lengths of the differentials are given by the differences of the exponents of theAlexander polynomial.Indeed, multiplying both sides by ( t p − t p +1 − t p − t p +1 − (cid:16) p − (cid:88) i =0 t pi − t p − (cid:88) i =0 t ( p +1) i (cid:17) = ( t p +1 − t p ( p − p − − t ( t p − t ( p +1)( p − p +1 − t p + p +1 − t p + p − t + 1= ( t p ( p +1) − t − (cid:3) Remark 6.8.
For the torus knot T , , i.e., the case where p = 2, we can check by hand that a ( T p,p +1 ) = 1 a ( T p,p +1 ) = p − , since ∆ T , ( t ) = t − t + 1. Remark 6.9.
More generally, for the torus knot T p,p +1 , the horizontal arrows increase in lengthby one at each “step”, from 1 to p −
1, and the vertical arrows decrease in length by one at each“step”, from p − HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 21
Lemma 6.10.
The iterated torus knot T , p,p +1 , p ≥ , is an L -space knot with Alexander polyno-mial ∆ T , p,p +1 ( t ) = k (cid:88) i =0 ( − i t n i , for a decreasing sequence of integers n > n > . . . > n k with n = p + pn = p + p − n = p − . In particular, a ( T , p,p +1 ) = 1 a ( T , p,p +1 ) = p. Proof.
The fact that T , p,p +1 is an L -space knot follows from [Hed09, Theorem 1.10] (cf. [Hom11b]),where Hedden gives sufficient conditions for the cable of an L -space knot to again be an L -spaceknot.The form of the Alexander polynomial follows from the formula for the Alexander polynomial ofthe cable of knot, i.e., ∆ T , p,p +1 ( t ) = ∆ T , ( t p ) · ∆ T p,p +1 ( t ) , and Lemma 6.7. More precisely, for p ≥ T , p,p +1 = ( t p − t p + 1)( t p − p − t p − p − + t p − p − t p − p − + lower order terms)= t p + p − t p + p − + t p − + lower order terms . The case p = 2 follows easily from the fact that∆ T , , ( t ) = t − t + t − t + 1 . (cid:3) Lemma 6.11.
For p ≥ , m ≥ p − p − , and m (cid:54) = 1 , the iterated torus knot T p,p +1;2 , m +1 is an L -space knot with Alexander polynomial ∆ T p,p +1;2 , m +1 ( t ) = k (cid:88) i =0 ( − i t n i , for a decreasing sequence of integers n > n > . . . > n k with n = 2 p − p + 2 mn = 2 p − p + 2 m − n = 2 p − p + 2 m. In particular, a ( T , p,p +1 ) = 1 a ( T , p,p +1 ) = 2 p − . Proof.
This iterated torus knot is an L -space knot by [Hed09, Theorem 1.10]. The form of theAlexander polynomial follows from the following facts:∆ T p,p +1;2 , m +1 ( t ) = ∆ T p,p +1 ( t ) · ∆ T , m +1 ( t ) , ∆ T , m +1 ( t ) = m (cid:88) i =0 ( − i t i , and Lemma 6.7. More precisely,∆ T p,p +1;2 , m +1 ( t ) = (cid:16) p − (cid:88) i =0 t pi − t p − (cid:88) i =0 t (2 p +2) i (cid:17)(cid:16) m (cid:88) i =0 t i − m − (cid:88) i =0 t i +1 (cid:17) = (cid:16) t p − p − t p − p − + p − (cid:88) i =0 t pi − t p − (cid:88) i =0 t (2 p +2) i (cid:17)(cid:16) m (cid:88) i =0 t i − m − (cid:88) i =0 t i +1 (cid:17) = t p − p +2 m − t p − p − − t p − p +2 m − + t p − p − + (cid:16) p − (cid:88) i =0 t pi − t p − (cid:88) i =0 t (2 p +2) i (cid:17)(cid:16) m (cid:88) i =0 t i − m − (cid:88) i =0 t i +1 (cid:17) = t p − p +2 m − t p − p − − t p − p +2 m − + t p − p − + (cid:0) t p − p +2 m + lower order terms (cid:1) = t p − p +2 m − t p − p +2 m − + t p − p +2 m + lower order terms , where the last equality follows from the hypothesis that m > p . (cid:3) Recall that D denotes the (positive, untwisted) Whitehead double of the right-handed trefoil. Lemma 6.12.
As elements of the group F , (cid:74) D (cid:75) = (cid:74) T , (cid:75) . Proof.
In [Hed07, Theorem 1.2], Hedden determines the Z -filtered chain homotopy type of (cid:92) CF K ofthe Whitehead double of K in terms of (cid:92) CF K ( K ). We can use this result to determine (cid:92) CF K ( D ),from which we will deduce the class (cid:74) D (cid:75) using rank and grading considerations.Using Hedden’s result, we see that (cid:92) CF K ( D, j ) (cid:39) F ⊕ F − j = 1 F − ⊕ F − j = 0 F − ⊕ F − j = − j denotes the Alexandergrading. Moreover, Hedden proves that every non-trivial differential on this complex lowers theAlexander grading by exactly one, which is sufficient to completely determine the Z -filtered chainhomotopy type of (cid:92) CF K ( D ). Note that τ ( D ) = 1.Let x be a generator of (cid:100) HF ( S ) ∼ = H ∗ ( C { i = 0 } ). Note that x necessarily is positioned at (0 , i, j )-plane. Then [ x ] must be zero in H ∗ ( C { j = 1 } ) since the homology of C { j = 1 } issupported in i -coordinate 2. By considering the support of (cid:92) CF K ( D ), we see that x is in the kernel HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 23 of ∂ horz , so in order to vanish in H ∗ ( C { j = 1 } ), it must be in the image of ∂ horz , i.e., there exists aclass, say y , positioned at (1 , ∂ horz y = x. The class [ y ] is equal to zero in H ∗ ( C { i = 1 } ) since the homology of C { i = 1 } is supported in j -coordinate 2. But y cannot be in the image of the differential on C { i = 1 } , since ∂ = 0, where ∂ is the differential on CF K ∞ , and ∂ horz y (cid:54) = 0. Hence, the boundary of y in C { i = 1 } must benon-zero; denote this boundary by z . Notice that z has ( i, j )-coordinates (1 , ∂ = 0 reasons, the boundary of z in C { j = 0 } must be zero, and by gradingconsiderations, z is not in the image of the differential on C { j = 0 } .The complex CF K ∞ ( − T , ) is generated over F [ U, U − ] by a, b, c, with the differential ∂a = b∂c = b, where the generators are have the following ( i, j )-coordinates: a (0 , b (0 , c (1 , . Then in the tensor product
CF K ∞ ( − T , ) ⊗ F [ U,U − ] CF K ∞ ( D )the generator az + by + cx is non-trivial in both vertical and horizontal homology. Indeed, it is clearly in the kernel of thevertical differential, and cannot be in the image of the vertical differential, since cx does not appearin the vertical boundary of any element. Similarly, it is in the kernel but not the image of thehorizontal differential.Thus, ε (cid:0) CF K ∞ ( − T , ) ⊗ F [ U,U − ] CF K ∞ ( D ) (cid:1) = 0 , as desired. (cid:3) We are now ready to prove Proposition 4.8, showing that we have the following relations in F ,where 0 < p < q : • (cid:74) T p,p +1 (cid:75) (cid:28) (cid:74) T q,q +1 (cid:75) • (cid:74) D p,p +1 (cid:75) (cid:28) (cid:74) D q,q +1 (cid:75) • (cid:74) T p,p +1 (cid:75) (cid:28) (cid:74) D p,p +1 (cid:75) • (cid:74) T p,p +1;2 , m +1 (cid:75) (cid:28) (cid:74) T q,q +1;2 , m +1 (cid:75) , for m ≥ q − q − Proof of Proposition 4.8.
The proposition is now an easy consequence of the preceding lemmas.We have from Lemma 6.7 that a ( T p,p +1 ) = 1 a ( T p,p +1 ) = p − . Now Lemma 6.4 states that if a ( J ) = a ( K ) and a ( J ) < a ( K ), then (cid:74) J (cid:75) (cid:28) (cid:74) K (cid:75) , implying that (cid:74) T p,p +1 (cid:75) (cid:28) (cid:74) T q,q +1 (cid:75) , which proves the first assertion in the proposition.From Lemma 6.12, we have that (cid:74) D (cid:75) = (cid:74) T , (cid:75) , and from Proposition 5.1 that (cid:74) D p,p +1 (cid:75) = (cid:74) T , p,p +1 (cid:75) . Hence by Lemmas 6.1 and 6.10, a ( D p,p +1 ) = 1 a ( D p,p +1 ) = p, so by Lemma 6.4, (cid:74) D p,p +1 (cid:75) (cid:28) (cid:74) D q,q +1 (cid:75) and (cid:74) T p,p +1 (cid:75) (cid:28) (cid:74) D p,p +1 (cid:75) . Finally, by Lemma 6.11, we have that a ( T p,p +1;2 , m +1 ) = 1 a ( T p,p +1;2 , m +1 ) = 2 p − , for p ≥ m ≥ p − p − m (cid:54) = 1, and so (cid:74) T p,p +1;2 , m +1 (cid:75) (cid:28) (cid:74) T q,q +1;2 , m +1 (cid:75) . This completes the proof of the proposition. (cid:3)
We conclude this paper by showing that our examples, { D p,p +1 − T p,p +1 } p ≥ , of smoothlyindependent, topologically slice knots are smoothly independent from the examples of Endo [End95]and Hedden-Kirk [HK10]. Recall that Endo’s examples are pretzel knots of the form K t = K ( − t − , t + 1 , t + 3) , t ≥ . In particular, they are of genus one. The examples of Hedden-Kirk are (positive, untwisted) White-head doubles of certain torus knots.
Proposition 6.13. If K is a knot of genus one and ε ( K ) = 1 , then either a ( K ) (cid:54) = 1 or a ( K ) = a ( K ) = 1 . Proof.
Notice that the assumption that ε ( K ) = 1 does not cause any loss of generality, since ε ( − K ) = − ε ( K ).Assume that a ( K ) = 1. We first notice that if K is a knot of genus one and ε ( K ) = 1, then τ ( K ) (cid:54) = −
1. This follows from the adjunction inequality for knot Floer homology [OS04, Theorem5.1], and the basis from Lemma 6.2Now, suppose a ( K ) = 1 and τ ( K ) = 0. Using the adjunction inequality [OS04, Theorem 5.1],and a basis found using the first part of Lemma 6.2, we see that the basis element x must be inthe kernel of the differential on C { i = 1 } . Moreover, for ∂ = 0 reasons, it cannot be in the imageof the differential on C { i = 1 } . But [ x ] cannot be zero in H ∗ ( C { i = 1 } , because τ ( K ) = 0 impliesthat H ∗ ( C { i = 1 } ) is supported in ( i, j )-coordinate (1 , a ( K ) = 1 and τ ( K ) = 1, in which case the arguments in the proofof Lemma 6.12 lead us to the desired result. (cid:3) HE KNOT FLOER COMPLEX AND THE SMOOTH CONCORDANCE GROUP 25
In the proof of Proposition 4.8, we showed that a ( D p,p +1 − T p,p +1 ) = 1 a ( D p,p +1 − T p,p +1 ) = p, Hence, by Proposition 6.13, along with Lemmas 6.3 and 6.4, it follows that when p >
1, ourexamples are independent from those of Endo and Hedden-Kirk.The following proposition describes the subgroup of F generated by Whitehead doubles: Proposition 6.14.
Whitehead doubles are contained in the rank one subgroup of F generated bythe right-handed trefoil.Proof. The argument in Lemma 6.12 can be used to show that for a Whitehead double
W D with ε ( W D ) = 1, the class (cid:74)
W D (cid:75) = (cid:74) T , (cid:75) in F . This is sufficient for the result, since ε ( W D ) = − ε ( − W D ) = 1, and ε ( W D ) = 0 implies that (cid:74)
W D (cid:75) = 0. (cid:3)
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