The anti-diagonal filtration: reduced theory and applications
aa r X i v : . [ m a t h . G T ] J un The anti-diagonal filtration: reduced theory and applications
Eamonn TweedyMay 29, 2018
Abstract
Given a knot K ⊂ S , Seidel and Smith described a graded cohomology group Kh symp,inv ( K ),a variant of their symplectic Khovanov cohomology group [SS10]. They also constructed a spec-tral sequence converging to the Heegaard Floer homology group d HF (Σ( K ) S × S )) with E -page isomorphic to a summand of Kh symp,inv ( K ). A previous paper [Twe14] showed thatthe higher pages of this spectral sequence are knot invariants. Here we discuss a reduced ver-sion of the spectral sequence which directly computes d HF (Σ( K )). Under some degenerationconditions, one obtains a new absolute Maslov grading on the group d HF (Σ( K )). This occurswhen K is a two-bridge knot, and we compute the grading in this case. We also extract some Q -valued knot invariants from this construction. INTRODUCTION
Let K ⊂ S be a knot, and let Σ( K ) denote the double cover of S branched along K . Thispaper is a continuation of a previous one [Twe14] in which we studied the invariance properties ofthe spectral sequence whose E page is isomorphic to a direct summand of a particular variant ofSeidel and Smith’s symplectic Khovanov cohomology and which converges to the Heegaard Floerhomology group d HF (Σ( K ) S × S )). In [Twe14], we proved that the filtered chain homotopytype of the filtered chain complex inducing this spectral sequence is a knot invariant; this impliesthat the higher pages of the spectral sequence are knot invariants also. The present paper will givea definition for a reduced version of the theory in the form of a filtration on the Heegaard Floerchain complex d CF (Σ( K )).Let b ∈ B n be a braid whose plat closure is a diagram of the knot K . Reviewing Manolescu’sconstruction in [Man06] and following the work of Bigelow [Big02], we described in [Twe14] howto define a fork diagram for b and compute a function R : G → Z + , where G is a set of Bigelowgenerators in the diagram. The notion of a fork diagram, as well as the definitions of the gradings T and Q are originally due to Bigelow [Big02]. As suggested in [Man06], one can define a reducedversion of R (denoted by R ) by considering a reducible fork diagram ; a set G of reduced Bigelowgenerators are determined by omitting a pair of arcs from the fork diagram in a prescribed way.We’ll review the notion of a fork diagram in more detail in Section 3 below, as well as describehow to reduce them and how to define reduced versions of the gradings P , Q , and T that appearin [Man06]. A certain holomorphic volume form is used in [Twe14] to define a grading e R on theunreduced Bigelow generators, and here we’ll use an analogous form to define a reduced version e R : G → Z . One can in fact compute this grading via the formula e R = T − Q + P , where the functions
Q, T , P : G → Z are analogous to their unreduced counterparts Q , T , and P found in [Twe14].We acquire R from e R via a rational shift s R ; let R = e R + s R ( b, D ) , where s R ( b, D ) = e ( b ) − w ( D ) − n − . These reduced Bigelow generators are in one-to-one correspondence with a set of generators for d CF (Σ( K )) (notice that the S × S summand found in the unreduced theory has been removed).Since Σ( K ) is a rational homology sphere, every s ∈ Spin c (Σ( K )) is torsion and so the entirecomplex d CF (Σ( K )) carries the Q -valued absolute grading e gr defined by Ozsv´ath and Szab´o in[OS06]. Thus we can define a filtration grading ρ on d CF (Σ( K )) via ρ = R − e gr Let P stand for either the ring of integers Z or for a field F . Now consider two distinguishedfiltered chain complexes V and W of free modules over P defined by V P ∗ := H ∗ (cid:0) S ; P (cid:1) and W P ∗ := H ∗ (cid:0) S ; P (cid:1) Now promote V and W to Z -filtered complexes by declaring each to supported in filtration level 0.2ne would hope to relate the reduced and unreduced theories. Observe that the functions( R +1 /
2) and R also provide filtrations on the Heegaard Floer complexes; Proposition 1.0.1 is provedin Section 3.9, and provides a correspondence between the reduced and unreduced complexes. Notethat we use ( R + 1 /
2) rather than R because R is ( Z + 1 / Proposition 1.0.1.
Let P either stand for Z or a field F . Let b ∈ B n be a braid which inducesa reducible fork diagram and whose closure is a diagram for the knot K . Let H (respectively H ) be the Heegaard diagram for Σ( K ) (respectively Σ( K ) S × S ) ) provided by Proposition3.5.1 below (respectively Proposition 4.2.1 from [Twe14]). Let s ∈ Spin c (Σ( K )) and let s ∈ Spin c ( S × S ) denote the torsion element. Equip d CF ( H , s s ; P ) with the ( R + 1 / -filtrationand equip d CF ( H , s ; P ) with the R -filtration. Let V P and W P be the filtered complexes definedabove. Then the filtered complexes d CF ( H , s s ; P ) and d CF ( H , s ; P ) ⊗ P V P have the same filtered chain homotopy type. Furthermore, equipping d CF ( H , s s ; P ) with the ρ -filtration and d CF ( H , s ; P ) with the ρ -filtration, d CF ( H , s s ; P ) and d CF ( H , s ; P ) ⊗ P W P have the same filtered chain homotopy type. One would like to use Proposition 1.0.1 in conjunction with Theorem 1.0.1 from [Twe14] (theinvariance result for the unreduced theory) to obtain an invariance result for the reduced theory.However, this requires recovering the filtered chain homotopy type of d CF ( H , s ; P ) from the filteredchain homotopy type of d CF ( H , s ; P ) ⊗ P V P (and from that of d CF ( H , s ; P ) ⊗ P W P ). The followingfact indicates that this is possible when P = F , a field. Proposition 1.0.2.
Let F be a field, and let C and C be finite-dimensional Z -graded, Z -filteredchain complexes of vector spaces over F , and let X denote either one of the filtered complexes V F or W F defined above. If C ⊗ F X is filtered chain homotopy equivalent to C ⊗ F X , then C is filteredchain homotopy equivalent to C . Along with Theorem 1.0.1 from [Twe14] and Proposition 1.0.1, Proposition 1.0.2 implies thefollowing:
Theorem 1.0.3.
Fix F to either stand for R or ρ , and let F be a field. Let the braids b ∈ B n and b ′ ∈ B m have plat closures which are both diagrams for the knot K , and assume that both inducereducible fork diagrams. Let H and H ′ be the pointed Heegaard diagrams for Σ( K ) induced by b and b ′ ,respectively, in the sense of Proposition 3.5.1 below. Then the F -filtered chain complexes d CF ( H ; F ) and d CF ( H ′ ; F ) have the same filtered chain homotopy type.Remark . Theorem 1.0.1 of [Twe14] was in fact only stated with respect to the filtration ρ .However, it is implicit in the proof of that result that an analogous invariance statement holds forthe filtration R as well. 3he filtration ρ induces a reduced version of the spectral sequence (over F ) with pages E k , andTheorem 1.0.3 implies the following. Corollary 1.0.5.
For k ≥ , the page E k is a knot invariant. One therefore obtains the following relationship between the pages of the spectral sequencesinduced by ρ and ρ , indicating that the reduced spectral sequence determines the unreduced one: Corollary 1.0.6.
For k ≥ , E k ∼ = E k ⊕ E k as Z -graded F -vector spaces. In Section 4.1.1, we show that the reduced theory enjoys a K¨unneth-type theorem with respectto connected sums of knots; this result provides a computational tool for composite knots.
Theorem 1.0.7.
Fix F to either stand for R or ρ , and let F be a field. Let K , K ⊂ S be knots,let s i ∈ Spin c (Σ( K i )) , i = 1 , . Then the filtered chain complexes d CF (Σ( K K ) , s s ; F ) and d CF (Σ( K ) , s ; F ) ⊗ F d CF (Σ( K ) , s ; F ) have the same filtered chain homotopy type, where d CF (Σ( K K ) , s s ; F ) is equipped with the F -filtration and d CF (Σ( K ) , s ; F ) ⊗ F d CF (Σ( K ) , s ; F ) is equipped with the tensor product filtrationinduced by the F -filtrations on the factors. Theorem 1.0.7 implies the following fact regarding the reduced spectral sequence:
Corollary 1.0.8.
For k ≥ , E k ( K K ) ∼ = E k ( K ) ⊗ F E k ( K ) . Given a knot K and some s ∈ Spin c (Σ( K )), we denote by d CF ∗ (Σ( K ) , s ) the dual complex to d CF ∗ (Σ( K ) , s ). Then d CF −∗ (Σ( K ) , s ) is a chain complex, and we define a filtration ρ ∗ via ρ ∗ ( x ∗ ) = − ρ ( x ) for each x ∈ T α ∩ T β . Theorem 1.0.9.
Fix F to either stand for R or ρ , and let F be a field. Let − K denote the mirrorimage of the knot K ⊂ S , and let s ∈ Spin c (Σ( K )) . Then d CF ∗ (Σ( − K ) , s ; F ) and d CF −∗ (Σ( K ) , s ; F ) are filtered chain isomorphic, where the complexes carry the filtrations ρ and ρ ∗ , respectively. Section 4.2 will discuss how one can distill a family of knot invariants from the reduced filtrationin the form of a function r K : Spin c (Σ( K ); F ) → Q . Restricting r K to the unique Spin structure s on Σ( K ), one obtains the Q -valued knot invariant r ( K ) = r K ( s ). Theorems 1.0.7 and 1.0.9 implythe following: Corollary 1.0.10.
Let K , K ⊂ S be knots, and let s i ∈ Spin c (Σ( K i )) for i = 1 , . Then r K K ( s s ) = r K ( s ) + r K ( s ) and r ( K K ) = r ( K ) + r ( K ) . Corollary 1.0.11.
Let K ⊂ S be a knot and let s ∈ Spin c (Σ( K )) . Then r − K ( s ) = − r K ( s ) and r ( − K ) = − r ( K ) .
4n [SS10], Seidel and Smith conjectured the existence of a concordance invariant arising fromthis theory. Motivated by their suggestion and by Corollaries 1.0.10 and 1.0.11, we make thefollowing speculation:
Conjecture 1.0.12.
Let C denote the smooth knot concordance group. The knot invariant r ( K ) provides a well-defined group homomorphism r : C → Q . We defined in [Twe14] the notion of ρ -degeneracy of a knot (see Definition 4.3.1 below). Thefollowing is a consequence of Proposition 1.0.1: Proposition 1.0.13.
Let K ⊂ S be a ρ -degenerate knot. Then the following hold:(i) The filtration on d HF (Σ( K ); F ) induced by R lifts the relative Maslov Z -grading on each non-trivial factor d HF (Σ( K ) , s ; F ) (ii) The grading R is an invariant of K . One sees this behavior when K is a two-bridge knot. Theorem 1.0.14.
Let F be a field. Let K ⊂ S be a two-bridge knot. Then K is ρ -degenerate, anddim F (cid:16) d HF R = k (Σ( K ); F ) (cid:17) = ( det ( K ) if k = σ ( K )2 otherwise , where σ ( K ) denotes the classical signature of K and det ( K ) denotes the determinant of K . We speculate the following, as suggested by Seidel and Smith in [SS10]:
Conjecture 1.0.15.
Every knot K ⊂ S is ρ -degenerate. The constructions in this paper and in [Twe14] can be expanded to links, with the limitationthat one can only define the filtrations ρ and ρ on the summands of the d CF complexes correspondingthe torsion Spin c -structures. We’ll make use of this extension in the in Section 3.9 - see Remark3.8.3. The situation for links is discussed further in Section 5. Remark . From now on, we’ll only explicitly include the coefficient ring in the notation whenthe distinction is necessary. The background constructions of Section 2 and constructions of Section3 through Section 3.9 work over either of Z or over F . In [OS04b], Ozsv´ath and Szab´o define the Heegaard Floer homology group d HF ( M ) associated toa connected, closed, oriented 3-manifold M . A genus-g Heegaard splitting for such a manifold canbe described via a pointed Heegaard diagram H αβ = (Σ; α ; β ; z ), where Σ is the splitting surface, α and β are g-tuples of attaching curves for the handlebodies, and z ∈ (Σ \ ∪ α i \ ∪ β i ). Recall thefollowing definitions: 5 efinition 2.0.17. Let (Σ; α ; β ; z ) be a pointed Heegaard diagram, and let D , . . . , D m be theconnected components of Σ \ ( ∪ α i ) \ ( ∪ β i ), where z ∈ D m . Then a two-chain P := m − X i =1 n i D i with n i ∈ Z is called a periodic domain if its boundary is a sum of α and β circles. Definition 2.0.18.
A Heegaard diagram (Σ; α ; β ; z ) is called admissible if every periodic domainhas both positive and negative coefficients.Given an admissible pointed Heegaard diagram one can compute the group d HF ( M ), which isthe Lagrangian Floer homology of the tori T α := α × . . . × α g and T β := β × . . . × β g lying insideof the symplectic manifold Sym g (Σ \ z ).More precisely, the group d CF ( H αβ ) is generated by the set of intersections T α ∩ T β ⊂ Sym g (Σ),and the differential is given by b ∂ ( x ) = X y ∈ T α ∩ T β X { φ ∈ π ( x , y ) | µ ( φ )=1 ,n z ( φ )=0 } (cid:16) c M ( φ ) (cid:17) y , where π ( x , y ) denotes the group of homotopy classes of 2-gons connecting x and y , c M ( φ ) denotesthe reduced moduli space of pseudo-holomorphic representatives for the class φ , µ ( φ ) denotes theMaslov index of φ , and n z ( φ ) := Im( φ ) ∩ (cid:0) { z } ∩ Sym g − (Σ) (cid:1) . Classes of such 2-gons are typically studied by analyzing their “shadows” in the surface Σ.
Definition 2.0.19.
Let (Σ; α ; β ; z ) be a pointed Heegaard diagram, and denote by D , D , . . . , D N the connected components of Σ \ ( ∪ i α i ) \ ( ∪ i β i ) , where D is the component containing the basepoint z . Then for 0 ≤ j ≤ N , choose a point z j in the interior of D j . For some class φ ∈ π ( x , y ) for x , y ∈ T α ∩ T β , the domain of φ is the 2-chain D ( φ ) := N X j =0 n j D j where n j := Im( φ ) ∩ (cid:0) { z j } × Sym g − (Σ) (cid:1) . We’ll say that φ avoids the basepoint if n = 0 (equivalently, n z ( φ ) = 0).Recall that there is a function s z : T α ∩ T β −→ Spin c ( M )partitioning T α ∩ T β into equivalence classes U s . This function induces decompositions d CF ( H ) = M s ∈ Spin c ( M ) d CF ( H , s ) and d HF ( M ) = M s ∈ Spin c ( M ) d HF ( M, s ) . For each s ∈ Spin c ( M ) the chain complex d CF ( M, s ) carries a relative grading gr defined via theMaslov index. For s ∈ Spin c ( M ) torsion, Ozsv´ath and Szab´o use surgery cobordisms to construct6n [OS06] an absolute Q -valued grading e gr on U s which lifts the relative grading in the followingsense: if x , y ∈ U s , then e gr ( x ) − e gr ( y ) = gr ( x , y ) . Whenever b ( M ) = 0, all Spin c structures on M are torsion and so the group d HF ( M ) can beabsolutely graded via e gr . In particular, this holds for M = Σ( K ) for a knot K ⊂ S . AlthoughSpin c (Σ( K ) S × S )) contains non-torsion elements, the group d HF (Σ( K ) S × S ) , s ) is non-trivial only if s is torsion.There is an analogous notion of a pointed Heegaard triple-diagram (resp. quadruple-diagram ),and one can study triply-periodic domains (resp. quadruply-periodic domains ) in such a diagram; apointed triple-diagram (resp. quadruple-diagram) is called admissible if every triply-periodic (resp.quadruply-periodic) domain has both positive and negative coefficients.A cobordism between closed 3-manifolds can be described be an admissible pointed triple-diagram (Σ; α ; β ; γ ; z ), which in turn induces a chain map b f αβγ : d CF ( H αβ ) ⊗ d CF ( H βγ ) → d CF ( H αγ ) given by b f αβγ ( x ⊗ y ) = X w ∈ T α ∩ T γ X { ψ ∈ π ( x , y , w ) | µ ( ψ )=0 ,n z ( ψ )=0 } ( M ( ψ )) w Here π ( x , y , w ) is the space of homotopy classes of 3-gons connecting x , y , and w , M ( ψ ) is themoduli space of pseudo-holomorphic representatives for the class ψ , and µ ( ψ ) is the Maslov indexof ψ .We’ll be particularly interested in maps induced by Heegaard moves. Definition 2.0.20.
Let (cid:0) Σ; α ; β ; β ′ ; z (cid:1) be a pointed Heegaard triple-diagram.(i) Let β ′ j differ from β j by an isotopy (avoiding z ) such that β ′ j intersects β j transversely intwo canceling points and β j ∩ β ′ i = ∅ when i = j. Then we say that β ′ differs from β by a pointed isotopy . A pointed isotopy which preserves the set of intersection points T α ∩ T β inthe obvious way will be called a small pointed isotopy .(ii) Instead let β , β , and β ′ bound an embedded pair of pants disjoint from z such that β ′ intersects β transversely in two points. Assume also that β j ∩ β ′ i = ∅ for i = j , and thatfor i > β ′ i relates to β i as ( i ) above. Then we say that β ′ differs from β by a pointedhandleslide .When a cobordism is induced by a pointed isotopy or a pointed handleslide relating two di-agrams for the same manifold, one can use the chain map described above to define a chain ho-motopy equivalence. More precisely, consider an admissible pointed Heegaard quadruple-diagram (cid:16) Σ; α ; β ; β ′ ; e β ; z (cid:17) such that β ′ differs from β by a pointed handeslide or pointed isotopy andsuch that e β differs from β by a small pointed isotopy. There is a distinguished representative θ ββ ′ ∈ T β ∩ T β ′ for the top-degree generator of d HF ( H ββ ′ ), and the 3-gon counting chain map b f αββ ′ ( · ⊗ θ ββ ′ ) : d CF ( H αβ ) → d CF ( H αβ ′ ) is a homotopy equivalence whose homotopy inverse isgiven by b f αβ ′ β ( · ⊗ θ β ′ β ) : d CF ( H αβ ′ ) → d CF ( H αβ ). The homotopies relating their compositions toidentity maps are constructed from maps counting holomorphic representatives of 4-gons arising inthe diagram (cid:16) Σ; α ; β ; β ′ ; e β ; z (cid:17) . 7n the other hand, if α ′ differs from α by a pointed handleslide or pointed isotopy and e α differsfrom α by a small pointed isotopy such that the pointed quadruple-diagram (Σ; e α ; α ′ ; α ; β ; z )is admissible, then b f α ′ αβ ( θ α ′ α ⊗ · ) is a homotopy equivalence with homotopy inverse given by b f αα ′ β ( θ αα ′ ⊗ · ). For a more detailed description of 3-gon counting chain maps and 4-gon countingchain homotopies, see the review in Section 2.1 of [Twe14] or the original discussion in Section 8of [OS04b].The following lemmas assure that certain pointed handleslides and pointed isotopies preserveadmissibility of pointed Heegaard diagrams. Lemma 2.0.21 is from [SW10], and the proof appearingthere involves a straightforward analysis of domain coefficients. We’ll use a similar method here toprove Lemma 2.0.22. Lemma 2.0.21. ([SW10]) Let H and H ′ be pointed Heegaard diagrams which differ in a localregion as shown in Figure 1, and coincide elsewhere. Then if H is admissible, so is H ′ . (a) Before the isotopy (b) After the isotopy Figure 1: A pointed isotopy
Lemma 2.0.22.
Let H and H ′ be two pointed Heegaard diagrams such that H can be obtained from H ′ by a handleslide of the form shown in Figure 2. Then if H is admissible, so is H ′ . D D D D D D D D (a) Before the handleslide D ′ D ′ D ′ D ′ D ′ D ′ D ′ D ′ e D e D e D e D e D (b) After the handleslide Figure 2: Heegaard diagrams before and after a certain pointed handeslide. The α arcs are solidand the β arcs are dotted (or vice versa). There can be arbitrarily many radial arcs on the annulusand arbitrarily many vertical arcs on the “neck” in the center. Proof.
Label the n (resp. n + 5) regions of the pointed Heegaard diagram H (resp. H ′ ) as indicatedin Figure 2(a) (resp. 2(b)), where D ′ k coincides with D k for k >
8. Now let P ′ be a periodic domainin H ′ with P ′ = b e D + b e D + b e D + b e D + b e D + n X j =1 c j D ′ j c − b = c − c = b − b = b − c = c − b = c − b and b − c = b − c = b − c .As a result, b = c , b = c , and c − c = c − c = c − c , and so there is a periodic domain P in H with P = n X j =1 c j D j . Since H is admissible, there are both positive and negative integers among the c j . d C F (Σ( K )) We describe how to define a reduced version of the ρ -filtration, denoted by ρ , which is a Q -valuedfiltration on the chain complex d CF (Σ( K )) (a definition first mentioned by Manolescu in [Man06]).This reduced version is much simpler to compute than the unreduced theory, and Theorem 1.0.3gives an invariance result for the reduced filtration.First we discuss some terminology which we’ll use when constructing the reduced filtration. Let B n denote the braid group on 2 n strands. This group is generated by { σ , . . . , σ n − } , where σ k denotes a half-twist of the k th strand over the ( k + 1) st strand. Given a braid b ∈ B n , we canobtain a diagram of a link called the plat closure of b by connecting ends of consecutive strandswith segments at the top and bottom, as shown in Figure 3.Figure 3: The left-handed trefoil is the plat closure of σ ∈ B . Clearly many pairs of braids have isotopic links as their plat closures; two braids have isotopicplat closures if and only if they can be related by a finite sequence of standard moves called
Birmanmoves ([Bir76]).
Let D ⊂ C denote the unit disk, choose 2 n points µ , . . . , µ n evenly spaced along R ∩ D , andlet the set of punctures be denoted by τ . We can view the braid group B n as the mapping classgroup of the punctured disk D n := D \ τ , where the generator σ k is a diffeomorphism which isthe identity outside of a neighborhood of the k th and ( k + 1) st punctures and exchanges these twopunctures by a counter-clockwise half-twist. Any braid can be written as a word in the σ k ’s, andwe view them as operating on D n in this way, read from left to right. Definition 3.2.1.
Let D ⊂ C be the unit disk and let b ∈ B n be an oriented braid on 2 n strands.9i) Let the standard fork diagram in D n be a collection of embeddings α , . . . , α n : I → D and h , . . . , h n : I → D called tine edges and handles , respectively, such that the following hold:(a) The arcs { α k ( I ) } nk =1 are pairwise disjoint horizontal segments and the arcs { h i } nk =1 arepairwise disjoint vertical segments.(b) For each k , we have that α k (0) = µ k − , α k (1) = µ k , h k (1) = d k ∈ ∂D, and h k (0) = m k = 12 ( µ k − + µ k ) . (ii) Let a fork diagram for b be the standard fork diagram along with the compositions b ◦ α , . . . , b ◦ α n and b ◦ h , . . . , b ◦ h n . We’ll also let β k := b ◦ α k .(iii) Let an augmented fork diagram for b be obtained from a fork diagram by replacing each arc β k with bE k , where E k is an immersed figure-eight which encircles µ k − and µ k and is orientedsuch that it winds counter-clockwise about µ k . (a) Tine edges α k (b) Figure-eights E k Figure 4: Structures in fork diagramsThe reader should notice that by drawing a picture containing just the α and β arcs and treatingthe α arcs as undercrossings at each intersection, we get a diagram of the plat closure of b .If the plat closure of the braid b is the knot K , then we will often refer to a fork diagram for b as a fork diagram for K .The reduced grading R is computed for a reduced set G of Bigelow generators given by omittinga pair of arcs α n , β n from the fork diagram (with a mild restriction on the diagram used). We’llsee that the set G is in one-to-one correspondence with a set of generators for d CF ( H ), where H is an admissible Heegaard diagram for the manifold Σ( K ) and is obtained from the reduced forkdiagram. We can define the reduced filtration for fork diagrams of a special type.
Definition 3.3.1. A reducible fork diagram for a knot K is a fork diagram for K with at least fourpunctures such that 10 µ n ∈ α n ∩ β n and • There exists an arc which avoids n − [ i =1 α i ! ∪ n − [ i =1 β i ! and connects µ n to a point on theboundary of the unit disk.Notice that a reducible fork diagram exists for any knot K , as one can be obtained by performinga Birman stabilization on any braid whose closure is K . We’ll define the reduced theory by omittingthe pair of arcs α n , β n from a reducible fork diagram for K (and subsequently omitting the figure-eight bE n from the augmented fork diagram).Consider a reducible fork diagram for K induced by a braid b ∈ B n . Denote by e Z the set ofintersections α i ∩ β j , where i, j ≤ n −
1. Similarly, define Z to be points α i ∩ bE j , i, j ≤ n −
1. Wethen define e G = ( α × . . . × α n − ) ∩ ( β × . . . × β n − ) ⊂ Conf n − ( C ) , G = ( α × . . . × α n − ) ∩ ( bE × . . . × bE n − ) ⊂ Conf n − ( C ) , where Conf k ( C ) denotes the k -fold configuration space of C , i.e. the set of unordered k -tuplesof distinct points in C . The relationship between bE k and β k induces a natural projection map p : Z → e Z . Letting τ denote the set containing the leftmost (2 n −
2) punctures, we see that τ ⊂ e Z . If x ∈ τ , p − ( x ) contains one point. If x ∈ Z \ τ , then p − ( x ) = { e x , e ′ x } , a two-pointset; we distinguish between e x and e ′ x by requiring that the loop traveling along a figure-eight from e x to e ′ x and back to e x along an α arc has winding number +1 around the puncture. Remark . Via an abuse of notation, we’ll often refer to the points corresponding to x ∈ e Z as x ∈ Z (if x ∈ τ ) or x, x ′ ∈ Z (if x ∈ e Z \ τ ). We define Z -valued functions Q , T , and P on G ; they only vary from their unreduced counterparts Q , T , and P found in [Twe14] in that the reduced versions ignore the omitted pair α n , β n . For aconcrete computation (albeit in the unreduced case), see Section 3.2 of [Twe14].We’ll first establish some notation that will be used throughout this section. Let x ∈ α i ∩ β j for 1 ≤ i, j ≤ n −
1. Let the loop γ x : I → D be the path from d j to d i formed by concatenatingthe following paths in the following order (here “ − γ ” denotes the reversal of a path γ ):(i) The path − bh j traveling from d j to bh j (0) ∈ β j (ii) The section of the path ± β j traveling from bh j (0) to x (iii) The section of the path ± α i traveling from x to h i (0) = m i (iv) The path h i traveling from m i to d i Furthermore, for a point e ∈ α i ∩ bE j , let e γ e denote the path from d j to d i formed analogously,except with the segment of ± β j replaced with a corresponding segment of the figure-eight ± bE j .Given a collection of paths γ , . . . , γ k : I → D with γ i ( t ) = γ j ( t ) for each t ∈ I if i = j ,the path ( γ , . . . , γ k ) : I → D × . . . × D descends to a path in Conf k ( D ) by composing withthe quotient map D × . . . × D . Futhermore, if the sets { γ (0) , . . . , γ k (0) } and { γ (1) , . . . , γ k (1) } k ( D ). Also recall that one can identify the braid group B k on k strands with the fundamental group of the k -fold configuration space Conf k ( D ). We’ll denote by π k : B k = π (cid:0) Conf k ( D ) (cid:1) → Z the usual abelianisation map. T Given a Bigelow generator e = e e . . . e n − ∈ G , we have that for each k , p ( e k ) = x k for some x k ∈ e Z . Now for each k , let γ k := γ x k be the path described above. Then let γ : I → Conf n − ( D )be the loop which is the image of ( γ , . . . , γ n − ) : I → D × . . . × D . We then define the grading T to be T ( e ) := π n − ( γ ) ∈ Z .There’s a convenient way to view T as a relative grading. Choose two reduced Bigelow generators e , e ∈ G with images x , x ∈ e G , and construct a loop in Conf n − ( D ) by traveling from x to x along the β curves and traveling from x back to x along the α curves. Then the difference T ( e ) − T ( e ) is the linking number of this loop with the fat diagonal ∇ ⊂ Sym n − ( D ). Inpractice, this is equal to the number of half-twists among the β -arcs connecting x to x . Thefunction T : G → Z is determined by this relative grading information along with the value of T ( e )for any one generator e ∈ G . Q Consider some e = e e . . . e n − ∈ G . For each j , let e γ k := e γ e k be the path defined as above.Then the path ( µ , . . . , µ n , e γ , . . . , e γ n − ) : I → D × . . . × D induces a loop in Conf n − ( D ) basedat { µ , . . . , µ n , d , . . . , d n − } . Note that the integer π n − ( e γ ) has the same parity as the integer π n − ( γ ) appearing above in the definition of T ( e ). We then define the Q -grading by Q ( e ) := 12 (cid:0) π n − ( e γ ) − π n − ( γ ) (cid:1) Z In practice, the function Q : G → Z can be computed additively from a function Q ∗ : Z → Z .Consider some x ∈ α i ∩ bE j (1 ≤ i, j ≤ n − δ x be the loop based at d i obtained byconcatenating the path e γ x (as above) with the segment of ∂D traveling from d j to d i . Now define Q ∗ ( x ) be the winding number of the loop δ x about the set of punctures { µ , . . . , µ n } . Then foreach e = e e . . . e n ∈ G , it is clear that Q ( e ) := n − X i =1 Q ∗ ( e i ) . P The function P will be computed additively from P ∗ : Z → Z , which measures twice the relativewinding number of the tangent vectors to the figure eights E ′ k at the points in Z .For x ∈ Z , where x ∈ α i ∩ bE j , we define P ∗ ( x ) in the following way. We view the arc bh j asbeing oriented downward at the point where it intersects ∂D . Let bE j have the orientation inducedby the orientation on E j in the standard fork diagram. Then we let P ∗ ( x ) be twice the windingnumber of the tangent vector relative to the downward-pointing tangent vector at the point h ′ j ∩ ∂D .In other words, if the tangent vector makes k counter-clockwise half-revolutions and m clockwisehalf-revolutions as we travel first along bh j from bh j (0) to bh j (1) then along bE j to x , then we12et P ∗ ( x ) = m − k . This number is an integer because we assume that at any point x ∈ Z , thefigure-eight intersects the α arc at a right angle.Then for e = e e . . . e n − ∈ G , we define P ( e ) = n − X i =1 P ∗ ( e i ) . d HF (Σ( K )) Now let P µ ∈ C [ t ] be the monic polynomial whose set of roots is τ , the set of punctures. We definean affine variety b S by b S = { ( u, z ) ∈ C : u + P µ ( z ) = 0 } ⊂ C . Also , for k = 1 , . . . , n −
1, define the subspaces b α k and b β k of b S by b α k = (cid:26) ( u, z ) ∈ C : z = α k ( t ) , for some t ∈ [0 , u = ± q − P µ ( z ) (cid:27) and b β k = (cid:26) ( u, z ) ∈ C : z = β k ( t ) , for some t ∈ [0 , u = ± q − P µ ( z ) (cid:27) . Now denote by T b α and T b β the totally real tori in Sym n − ( b S ) defined by T b α = b α × . . . × b α n − and T b β = b β × . . . × b β n − . The map b S → C given by ( u, z ) z is in fact a 2-fold branched covering map, where the branch setdownstairs is the set of punctures τ . Thus (by viewing S as C ∪ { + ∞} ), one can see that in fact b S = Σ n − \{±∞} , where Σ n − is a closed surface of genus n −
1. In Section 4.2 of [Twe14] (following[Man06]), we described the construction of a pointed Heegaard diagram for Σ( K ) S × S ) in theunreduced setting - the extra S × S summand appeared because one was required to stabilize theHeegaard surface. Since we’ve omitted a pair of curves here, we don’t need to stabilize. Proposition 3.5.1.
The collection of data H = (Σ n − ; b α , . . . , b α n − ; b β , . . . , b β n − ; + ∞ ) is an admissible pointed Heegaard diagram for Σ( K ) .Proof. We first check that H actually represents Σ( K ). It suffices to show that H ′ = (cid:16) Σ n = Σ n − ; b α , . . . , b α n − , α ; b β , . . . , b β n − , β ; + ∞ (cid:17) gives a pointed Heegaard diagram for Σ( K ) S × S ), where (Σ ; α ; β ; + ∞ ) is the standardpointed Heegaard diagram for S × S shown in Figure 5. We accomplish this by showing that H can be obtained from H ′ by a sequence of handleslides, where H = (cid:16) Σ n ; b α , . . . , b α n ; b β , . . . , b β n ; + ∞ (cid:17) β aa + ∞ Figure 5: A pointed Heegaard diagram for S × S is the pointed Heegaard diagram for Σ( K ) S × S ) obtained as the stabilized double branchedcover of the unreduced fork diagram as in [Twe14].The set of attaching circles b α , . . . , b α n − , α can be obtained via a sequence of handleslides andisotopies (avoiding the basepoint + ∞ ) from the set b α , . . . , b α n . Indeed, one should slide b α n over b α n − , then slide the resulting curve over b α n − , and so on (for a total of n − b α , . . . , b α n − , α . This process is illustrated in Figure 6. As a result, β i is obtainedfrom α i (for each i ) via the diffeomorphism induced by the braid, b β , . . . , b β n can be obtained viaan analogous sequence of handleslides and isotopies from b β , . . . , b β n − , β . a b c b α b α b α a b c (a) a b ca b c (b) a b ca b cα (c) Figure 6: Handleslides connecting the set { b α , b α , b α } to { b α , b α , α } in the proof of Proposition3.5.1. Pairs of pants are shaded.Admissibility of the diagram H ′ follows by mimicking the argument in the proof of Proposition7.4 from [Man06]; when examining a periodic domain P , one exploits the involution on the regionsof Σ ( n − \ ( n − [ i =1 b α i \ ( n − [ i =1 b β i (induced by the involution on the Heegaard surface) to findpositive and negative coefficients in the expansion of P . Remark . With respect to the branched covering map b S → C , each puncture µ k ∈ C has asingle point as its preimage. However, the preimage of a point x ∈ β j ∩ int( α i ) consists of a pairof points upstairs. This suggests an identification between the intersection T b α ∩ T b β and the set G .However, this identification isn’t canonical, since for some x ∈ e Z \ τ it is only required that the pair { e x , e ′ x } is identified with the two preimages of x upstairs. Still, it is straightforward to check that (cid:0) P − Q (cid:1) and T descend to functions e G → Z and thus give well-defined functions T b α ∩ T b β → Z .14 .6 Gradings on reduced tori Recall that a system of coordinates on Sym ( n − ( b S ) is given by (cid:8) ( u , z ) , ( u , z ) , . . . , ( u ( n − , z ( n − ) (cid:9) where u i + P µ ( z i ) = 0. Define the anti-diagonal by the submanifold ∇ ⊂ Sym ( n − ( b S ) given by ∇ := {{ ( u i , z i ) i } where z j = z k and u j = − u k for some j = k } , and let W = Sym ( n − ( b S ) \ ∇ . One can lift the relative Maslov grading on T b α ∩ T b β ⊂ W to anabsolute Z -valued grading. This lift is achieved by first choosing a particular C -valued holomorphicvolume form on W and using this volume form to improve T b α and T b β to graded submanifolds .From this pair of graded submanifolds, one obtains an absolute Maslov grading on the set T b α ∩ T b β .This construction was introduced by Seidel in [Sei00] and reviewed in Section 4.2 of [Twe14]. Bycomparing this situation to that in the unreduced case, we obtain the following. Proposition 3.6.1.
There exists a volume form Θ on W inducing Seidel gradings on T b α and T b β in the sense of Section 4.1 of [Twe14], and the resulting absolute Maslov grading on T b α ∩ T b β ⊂ W is exactly P − Q + T .Proof. Consider the C -valued form Θ on Sym ( n − ( b S ) given byΘ := Y ≤ i We dispense with a type of 3-gon class which will arise often. Assume that α , β , and β ′ be n-tuplesof attaching curves on Σ such that β ′ differs from β by a pointed handeslide or pointed isotopy.For x ∈ T α ∩ T β and y ∈ T α ∩ T β ′ , let ψ ∈ π ( x , θ β , β ′ , y ) be a 3-gon class avoiding the basepointwith µ ( ψ ) = 0, where the domain D ( ψ ) is a sum of n disjoint 3-sided regions D , . . . , D n . A pointin Im( ψ ) ⊂ Sym n (Σ) is of the form x = { x , . . . , x n } , where each x i ∈ D i . If we further assumethat at least n − D i are thin 3-sided regions of the type in Figure 8, then π − ( π ( D i )will be the disjoint union of two thin 3-sided regions and so D i ∩ π − ( π ( D j )) = ∅ for i = j. As a result, Im( ψ ) ∩ ∇ = ∅ . θ ββ ′ x i y i Figure 8: A small 3-sided region appearing in a small iso-topy 17 .8 Triangle injections We define some auxiliary maps which will be useful for comparing ρ -filtered chain homotopy typesCHDs which cover fork diagrams. Definition 3.8.1. Let (Σ; α ; β ; z ± ; π ) and (cid:0) Σ; α ′ ; β ′ ; z ± ; π (cid:1) be two admissible CHDs of genus n appearing in some sequence of pointed isotopies and handleslides connecting two CHDs which coverfork diagrams, and let ∇ ⊂ Sym n (Σ) denote the anti-diagonal.(i) If β ′ = β and α ′ differs from α by a pointed isotopy or pointed handleslide, then a α -triangleinjection is a function g : T α ∩ T β ֒ → T α ′ ∩ T β such that the following hold:(a) There is an admissible triple-CHD (Σ; α + ; α ; β ; z ± ; π ) (where α + differs from α ′ by asmall pointed isotopy and for each k , α + k intersects α k transversely in two points) suchthat for each x ∈ T α ∩ T β , there is a 3-gon class ψ + g ∈ π ( θ α + α , x , y + ) with µ ( ψ + g ) = 0, ψ + g ∩ ∇ = ∅ , and n z ( ψ + g ) = 0, where y + ∈ T α ∩ T α + is the nearest neighbor to g ( x ).(b) There is an admissible triple-CHD (Σ; α ; α − ; β ; z ± ; π ) (where for each k , α − k differs from α ′ k by a small pointed isotopy and for each k , α − k intersects α k transversely in two points)such that for each x ∈ T ′ α ∩ T β , there is a 3-gon class ψ − g ∈ π ( θ αα − , y − , x ) with µ ( ψ − g ) = 0, ψ − g ∩ ∇ = ∅ , and n z ( ψ − g ) = 0, where y − ∈ T α ∩ T α − is the nearest neighborto g ( x ).(ii) If α ′ = α and β ′ differs from β by a pointed isotopy or pointed handleslide, then a β -triangleinjection is a function g : T α ∩ T β ֒ → T α ∩ T β ′ such that the following hold:(a) There is an admissible triple-CHD (cid:0) Σ; α ; β ; β + ; z ± ; π (cid:1) (where β + differs from β ′ by asmall pointed isotopy and for each k , β + k intersects β k transversely in two points) suchthat for each x ∈ T α ∩ T β , there is a 3-gon class ψ + g ∈ π ( x , θ ββ + , y + ) with µ ( ψ + g ) = 0, ψ + g ∩ ∇ = ∅ , and n z ( ψ + g ) = 0, where y + ∈ T β ∩ T β + is the nearest neighbor to g ( x ).(b) There is an admissible triple-CHD (cid:0) Σ; α ; β − ; β ; z ± ; π (cid:1) (where β − differs from β ′ by asmall pointed isotopy and for each k , β − k intersects β k transversely in two points) suchthat for each x ∈ T α ∩ T ′ β , there is a 3-gon class ψ − g ∈ π ( y − , θ β − β , x ) with µ ( ψ − g ) = 0, ψ − g ∩ ∇ = ∅ , and n z ( ψ − g ) = 0, where y − ∈ T β ∩ T β − is the nearest neighbor to g ( x ).Although the chain maps induced by moves on CHDs count holomorphic representatives of 3-gon classes, one can understand how they interact with the filtration ρ by studying the appropriatetriangle injections - this will allow us to avoid a difficult direct analysis of moduli spaces of 3-gons.This was accomplished via Lemma 5.0.6 in [Twe14]; the following is a modification of that result. Lemma 3.8.2. Let K ⊂ S be a knot and let H = (Σ; α ; β ; z ± ; π ) and H ′ = (cid:0) Σ; α ′ ; β ′ ; z ± ; π (cid:1) betwo CHDs for Σ( K ) which are obtained from braids b and b ′ (possibly after Heegaard stabilization),and suppose that the following hold:(i) For integers m, n with m ≥ n ≥ , there is a sequence of pointed isotopies and/or handleslides H = H 7→ H . . . 7→ H n 7→ H n +1 . . . 7→ H m = H ′ such that each intermediate CHD H i = (cid:0) Σ; α i ; β i ; z ± ; π (cid:1) is admissible. Recall that we use afixed projection map π : Σ → S throughout this sequence. ii) There are triangle injections g i : T α i − ∩ T β i − → T α i ∩ T β i (for ≤ i ≤ n ) and g i : T α i ∩ T β i → T α i − ∩ T β i − (for n < i ≤ m ) satisfyingIm ( g n ◦ g n − ◦ . . . ◦ g ) ⊆ Im ( g n +1 ◦ g n +2 ◦ . . . ◦ g m ) (iii) The composition g : T α ∩ T β → T α ′ ∩ T β ′ given by g := ( g n +1 ◦ g n +2 ◦ . . . ◦ g m ) − ◦ ( g n ◦ g n − ◦ . . . ◦ g ) satisfies R ( g ( x )) = R ( x ) for each x ∈ T α ∩ T β .Then for each s ∈ Spin c (Σ( K )) , the F -filtered complexes d CF ( H , s ) and d CF ( H ′ , s ) have the samefiltered chain homotopy type, where F can be chosen to be either R or ρ .Proof. The case in which m = n (i.e. H n = H ′ ) is very similar to that appearing in Lemma 5.0.6of [Twe14] - the statement there concerned the unreduced filtrations R and ρ and the manifoldΣ( K ) S × S ). Due to the similarities between the reduced and unreduced constructions, thesame argument works here as well.An analogous argument works when m > n , though the triangle injection g i “goes the wrongway” in the sequence of Heegaard moves when n < i ≤ m . To remedy this, one simply reverses theroles of the 3-gons ψ ± g i associated to g i for n < i ≤ m when running the argument from [Twe14].It is implicit in the proof of Lemma 5.0.6 of [Twe14] that under these hypotheses, the R -filteredcomplexes also have the same filtered chain homotopy type. Remark . Notice that if the plat closure of a braid b is a link L ⊂ S (rather than a knot),versions of Propositions 3.5.1 and 3.6.1 still hold and provide a filtration R on d CF (Σ( L )). However,the absolute grading e gr (and thus the filtration ρ ) is only defined for the summands d CF (Σ( L ) , s )with s ∈ Spin c (Σ( L )) torsion.Assuming one restricts to these torsion summands, an analogue of Lemma 3.8.2 holds for linksas well (the proof doesn’t rely on the number of components); this will be applied to a particulartwo-component link L K in the proof of Proposition 1.0.1 below. Section 5 will discuss the situationfor links in more generality. Proposition 1.0.1 provides the relationship between the filtrations ρ and ρ . However, we’ll needLemma 3.9.2, a technical fact which concerns a braid b ∈ B n − inducing a reducible fork diagramwhich looks as in Figure 9 in a neighborhood of the two rightmost punctures. Such a braid can beobtained for any knot via some appropriate stabilization. ...... Figure 9: The local configuration near µ n − and µ n − in Lemma 3.9.2. The dashed arc is β n − .Given a knot K , let L K be the two-component split link whose components are K and anunknot; recall that Σ( L K ) ∼ = Σ( K ) S × S ). As mentioned in Remark 3.8.3, one can endow19 CF (Σ( K ) S × S ) , s s ) with the R and ρ filtrations for s ∈ Spin c (Σ( K )) and s ∈ Spin c ( S × S )torsion.Now suppose that a fork diagram has some puncture µ k ∈ α j ∩ β j (for arbitrary j and k = 2 j − k = 2 j ) and an arc avoiding the other α i and β i and connecting µ k to the boundary of the unitdisk. Then one can define versions of R and ρ in the obvious way by omitting α j , β j (this is slightlymore general than the usual reduction procedure described immediately after Definition 3.3.1). Remark . From this point forward, all sequences of isotopies and handleslides passing throughCHDs will preserve the points z ± and the branched covering map π . Therefore, we’ll drop thesefrom the notation (though we’ll often keep track of the basepoint z + for Heegaard Floer purposes.Furthermore, all Heegaard diagrams should be understood to be CHDs (and we’ll often refer tothem as “Heegaard diagrams”). Lemma 3.9.2. Let b ∈ B n − be a braid inducing a fork diagram of the form shown in Figure9. Denote by H n the Heegaard diagram obtained by omitting the pair α n , β n from the reduciblefork diagram induced by the braid b × (1) ∈ B n , and by H n − theHeegaard diagram obtained byomitting the pair α n − , β n − from the same fork diagram. Then d CF ( H n ) and d CF ( H n − ) have thesame R -filtered chain homotopy type.Proof. We compare the filtered chain homotopy types of the complexes obtained from H n and H n − . This will be accomplished via an intermediate picture: let α ′ n be the arc obtained from α n via the finger-like isotopy shown in Figure 10, and let β ′ n = bα ′ n . ... Figure 10: The α -arcs in the forkdiagram F ′ ; α n is dashed, α ′ n isdotted, and α i is solid for i < n .Let F denote the reduced fork diagram obtained by omitting the pair α n , β n , and let F ′ denotethe diagram obtained from F by replacing α n − and β n − with α ′ n and β ′ n . We’ll denote by H ′ n − the diagram for Σ( K ) covering F ′ . Furthermore, let Z = ( α × . . . × α n − ) ∩ ( β × . . . × β n − ) and Z ′ = (cid:0) α × . . . × α n − × α ′ n (cid:1) ∩ (cid:0) β × . . . × β n − × β ′ n (cid:1) . ... ... aa bb cc dd Figure 11: The α -circles inthe Heegaard diagram H ′ n − The set of attaching circles α ′ = { b α , . . . , b α n − , b α ′ n } can be obtained from α = { b α , . . . , b α n − } via a sequence of n − n = 5). Let α := b α n − . For 1 ≤ i ≤ n − 3, the i th handleslide consists of sliding α i over b α n − ( i +1) to obtain α ( i + 1). Notice that we choose α to be the member of its isotopy class thathas the “fingers” depicted in Figure 12(a). For 2 ≤ i ≤ n − 3, we choose α i +1 to be the member ofits isotopy class which is very close to α i , as seen in Figure 12(b). In the final handleslide, we slide α n − over b α to obtain α n − , choosing the representative of its isotopy class that is depicted inFigure 12(c). Notice that the curve is the result of applying two “finger move” isotopies to the circlewhich is precisely the cover of the right-most arc α n that we omitted to reduce the fork diagram. a b c da b c d (a) a b c da b c d (b) a b c da b c d (c) Figure 12: Handleslides connecting the set { b α , b α , α := b α } to { b α , b α , α := b α ′ } . In each picture,the old curve α i is dashed, the new curve α i +1 is dashed, the unaffected b α k are solid, and the pairof pants is dashed.As the diffeomorphism induced by the braid carries b α i to b β i , the set of circles β ′ = { b β , . . . , b β n − , b β ′ n } can be obtained from the set β = { b β , . . . , b β n − } via an analogous sequence of n − i th handleslide is denoted by β i +1 and where β := b β n − . These handeslides are obtained by precomposing the α -handleslides with the braiddiffeomorphism. Note that the parallel α -curves in these handleslides (and thus the β curves whichare their images under the braid diffeomorphism) should be very close together - they’re shownsomewhat separated in Figure 12 for readability.Now we describe the sequence of Heegaard diagrams that we’ll study. Define the sets of attachingcurves α i , β i (for 1 ≤ i ≤ n − α i := (cid:8)b α , . . . , b α n − , α i (cid:9) and β i := n b β , . . . , b β n − , β i o Then for 1 ≤ i, j ≤ n − 1, define H i,j := (cid:0) Σ; α i ; β j ; + ∞ (cid:1) . Then we study the sequence of21andleslides given by H , 7→ H , . . . 7→ H n − , 7→ H n − , . . . 7→ H n − ,n − 7→ H n − ,n − 7→ H n − ,n − . Notice that H n = H , and H ′ n − = H n − ,n − . Recall that because they cover fork diagrams, H n and H n − are admissible. We should address admissibility of intermediate diagrams.We claim that the move connecting H , (resp. H n − , ) to H , (resp. H n − , ) is a composition ofan admissibility-preserving pointed handleslide and several admissibility-preserving pointed fingerisotopies. To use Lemma 2.0.22, we must exhibit an annular region of the diagram of the form ofthe region in the hypothesis of the Lemma. Figure 13 illustrates this annulus in a local picture ofthe handleslide H i, → H i +1 , for 2 ≤ i ≤ n − 3, and so those handleslides preserve admissibility.The required annular region for H n − ,i +1 → H n − ,i +1 can be obtained by transforming Figure 13by the braid diffeomorphism. The handleslides H , → H , can be realized as the compositionof a handleslide like the one in Figure 13 followed by a pair of finger isotopies, which preserveadmissibility by Lemma 2.0.21. Notice also that one can pass from H n − to H ′ n − to H n − ,n − to H n − ,n − via several admissibility-preserving pointed finger isotopies. aa (a) α i − and b α n − i aa (b) α i and b α n − i Figure 13: A local picture of thehandleslide of α i − (solid) over b α n − i (dashed) to yield α i (solid) appearing inthe proof of Lemma 3.9.2. The annulusof Lemma 2.0.22 is shaded.In Section 3.9.1 below, we’ll construct triangle injections g iα , g iβ for 1 ≤ i ≤ n − 3, where g iα : T α i ∩ T β → T α i +1 ∩ T β and g iβ : T α n − ∩ T β i → T α n − ∩ T β i +1 as well as triangle injections g n − α : T α n − ∩ T β n − → T α n − ∩ T β n − and g n − β : T α n − ∩ T β n − → T α n − ∩ T β n − Additionally, we’ll have thatIm (cid:16) g n − β ◦ . . . ◦ g β ◦ g n − α ◦ . . . ◦ g α (cid:17) ⊆ Im (cid:16) g n − α ◦ g n − β (cid:17) , which gives a function g = (cid:18)(cid:16) g n − α ◦ g n − β (cid:17) − ◦ g n − β ◦ . . . ◦ g β ◦ g n − α ◦ . . . ◦ g α (cid:19) : T α ∩ T β → T α ′ ∩ T β ′ By the version of Lemma 3.8.2 for links (see Remark 3.8.3), it then suffices to check that R ( g ( x )) = R ( x ) for all x ∈ T α ∩ T β - this is also checked in Section 3.9.1 below. Remark . One might ask whether it is possible to reduce a fork diagram by deleting any pair α k , β k , assuming that they satisfy technical conditions analogous to those found in Definition 3.3.1.This can be done, and the methods used in the proof of Lemma 3.9.2 could be extended to showthat the filtered chain homotopy type doesn’t depend on the choice of pair to delete. However, wedon’t prove this fact here, as it isn’t necessary for proving Proposition 1.0.1.22 roof of Proposition 1.0.1. Let b ∈ B n − be a braid whose plat closure is the knot K (with n > b ′ = b × (1) ∈ B n is the two-component link L K ; recall that Σ( L K ) ∼ = Σ( K ) S × S ). Let H n and H n − be the Heegaard diagrams for Σ( L K ) obtained from b ′ as in the statement of Lemma3.9.2.Let s denote the torsion Spin c -structure on S × S , and let s ∈ Spin c (Σ( K )). Equippedwith their respective R filtrations, Lemma 3.9.2 provides that the complexes d CF ( H n − , s s ) and d CF ( H n , s s ) have the same filtered chain homotopy type.Further, let H ′ denote the genus-( n − 1) Heegaard diagram for Σ( K ) S × S ) obtained from b in the unreduced sense. One can see that this diagram is isotopic to H n . Also, the fork diagramcovered by H n only differs from the fork diagram covered by H ′ by an extra pair of punctureswhich are far from any arcs in the diagram. Identifying Bigelow generators in the obvious wayand observing that s R ( b ) = s R ( b ′ ) , one obtains that the R -filtered complex d CF ( H ′ ) has the samefiltered chain homotopy type as the R -filtered complex d CF ( H n ).Now denote by H the genus-( n − 2) Heegaard diagram for Σ( K ) obtained from b in the reducedsense by omitting the pair α n − , β n − . Letting H denote the standard Heegaard diagram for S × S of genus 1 (containing only generators from s ), one can see that H n − = H H (wherethe sum region is near + ∞ ). As a result, all generators of d CF ( H n − ) lie in Spin c structures of theform s s . Further, the connected sum provides a natural correspondence between each generator x for d CF ( H , s ) and a pair of generators x y and x z for d CF ( H n − , s s ), where y, z ∈ b α n ∩ b β n . Nowthe fork diagram covered by H n − is obtained from the one covered by H via a connected sum with F , the two-puncture fork diagram for 1 ∈ B . One can then verify directly that P ∗ ( y ) = 2 , P ∗ ( z ) = 0 , Q ∗ ( y ) = 1 , Q ∗ ( z ) = 0 ,T ( x y ) = T ( x ) , T ( x z ) = T ( x ) , and s R ( b ′ ) = s R ( b ) − , and thus R ( x y ) = R ( x ) + 12 and R ( x z ) = R ( x ) − d CF ( H n − ) and d CF ( H ) with their respective R -filtrations, then thefiltered complex d CF ( H n − ) has the same filtered chain homotopy type as the filtered complex d CF ( H ) ⊗ V. Furthermore, notice that e gr ( x y ) = e gr ( x ) + 12 and e gr ( z ) = e gr ( x ) − , and so d CF ( H n − ) and d CF ( H ) ⊗ W have the same ρ -filtered chain homotopy types. The resultsfollow.In light of this relationship, one can now obtain an invariance result for the reduced theory. Proof of Theorem 1.0.3. Let H and H be two Heegaard diagrams for Σ( K ) obtained from re-ducible fork diagrams in the usual way, and let H H be corresponding diagrams for Σ( K ) S × S ) obtained from unreduced fork diagrams. Either fix F to stand for R and F to stand for R , orfix F to stand for ρ and F to stand for ρ . For s ∈ Spin c (Σ( K )) and s ∈ Spin c ( S × S ) torsion,equip d CF ( H i , s ; F ) with F filtrations and equip d CF ( H i , s s ; F ) with F filtrations. Proposition1.0.2 applies and the result follows via Proposition 1.0.1 and Theorem 1 from [Twe14].23 .9.1 Triangle injections for Lemma 3.9.2 We’ll define the sequence of triangle injections required for the proof of Lemma 3.9.2, exhibit therequired 3-gons, and check that the composition g preserves R .We first define the triangle injection g α : T α ∩ T β → T α ∩ T β . Consider some w x ∈ T α ∩ T β ,where x ∈ b α n − ∩ b β j , for 1 ≤ j ≤ n − 1. Then let g α ( w x ) = w y , where y ∈ α ∩ b β j is as indicatedin Figure 14. xya ba b (a) x ya ba b (b) Figure 14: Two possibilities for inter-sections x ∈ b α n − ∩ b β j and y ∈ α ∩ b β j appearing in the definition of the trian-gle injection g α . In each picture, b β j issolid, α = b α n − is dashed, and α isdotted.We illustrate the 3-gons ψ , + α and ψ , − α associated to g α . Recall from Definition 3.8.1 that weseek ψ , + α ∈ π ( θ α + α , w x, w y + ) and ψ , − α ∈ π ( θ α α − , w y − , w y ), where w ± y ± ∈ T α ± ∩ T β and α ± are small isotopic pushoffs of α with corresponding curves intersecting transversely in pairsof points. Components involving x appear in Figure 15. All other components are “small” 3-sidedregions as seen in Figure 16. x y + (a) ψ , + α xy + (b) ψ , + α x y − (c) ψ , − α xy − (d) ψ , − α Figure 15: Some components of 3-gons associated to g α . Parts (a) and (b) show possible com-ponents of ψ α and (c) and (d) show possible components of ψ − α . The dashed curve is b α n − andthe dotted curve is ( α ) + in (a) and (b) and ( α ) − in (c) and (d). In each picture, the white dotrepresents a component of the appropriate top-degree θ -generator.Now for 2 ≤ i ≤ n − 3, we define g iα : T α i ∩ T β → T α i +1 ∩ T β , the triangle injection associatedto the handleslide α i → α i +1 . Consider w x ∈ T α i ∩ T β , where x ∈ α i ∩ b β j for 1 ≤ j ≤ n − g iα ( w x ) = w y , where y ∈ α i +1 ∩ b β j is the obvious point close to x . The 3-gon componentsinvolving x are shown in Figure 17; all other components are as in Figure 16.The triangle injection g β : T α n − ∩ T β → T α n − ∩ T β is defined in a way that is somewhatanalogous to g α . For w x ∈ T α n − ∩ T β with x ∈ α ∩ b β n − (where α is some curve in the tuple α n − ), we define g β ( w x ) = w y , where y ∈ α ∩ β is as indicated in Figure 18.24 y + (a) ψ , + α x y − (b) ψ , − α Figure 16: Small components of 3-gons associatedto g α xy + (a) ψ i, + α xy − (b) ψ i, − α Figure 17: Some components of 3-gons associated to g iα with 2 ≤ i ≤ n − . In each picture, the leftmostdashed curve is b α n − i − . Figure 19 illustrates the component involving x for each 3-gon ψ , ± β associated to g β . Othercomponents are as in Figure 16. The definitions of g iβ for 2 ≤ i ≤ n − g iα , with associated 3-gons as in Figures 16 and 17.Notice that the last two handleslides α n − α n − and β n − β n − are inverses of han-dleslides which are of the form depicted in Figure 2. The injections g n − α : T α n − ∩ T β n − → T α n − ∩ T β n − and g n − β : T α n − ∩ T β n − → T α n − ∩ T β n − can thus be defined in a way that isanalogous to the above definition of g iα for 2 ≤ i ≤ n − (cid:16) g n − β ◦ . . . ◦ g β ◦ g n − α ◦ . . . ◦ g α (cid:17) ⊆ Im (cid:16) g n − α ◦ g n − β (cid:17) . As a result, we can define g = (cid:18)(cid:16) g n − α ◦ g n − β (cid:17) − ◦ g n − β ◦ . . . ◦ g β ◦ g n − α ◦ . . . ◦ g α (cid:19) : T α ∩ T β → T α ′ ∩ T β ′ . We’ll break out analysis of gradings into two cases, depending on the form of the initial element of Z . First consider w x ∈ Z , where x ∈ α n − ∩ β n − and w = { w , . . . , w n − } . Local pictures of F and F ′ can be seen in Figures 20 and Figure 21, respectively.Figure 22 shows local regions of the diagram H , (resp. H n,n ) covering a neighborhood of thepuncture µ n − in the fork diagrams F (resp. F ′ ). It is easily verified that either g ( w x ) = w z or g ( w x ) = w z ′ .However, we should justify that R ( w z ) = R ( w x ). Strictly speaking, we should calculate thegradings for w z after performing an isotopy on F ′ so that α ′ n is a horizontal joining µ n − and µ n ;the result is shown in Figure 23.One can verify using the fork diagrams in Figured 20 and 23 that indeed Q ( w z ) = Q ( w x ), P ( w z ) = P ( w x ), and T ( w z ) = T ( w x ). So, g preserves R in this case.25 bxya b (a) a bxya b (b) Figure 18: Two possibilities for intersections x ∈ α ∩ β and y ∈ α ∩ β appearing in the definitionof the triangle injection g β . In each picture, α is solid, β = b β n − is dashed, and β is dotted. x y + (a) ψ , + β xy + (b) ψ , + β x y − (c) ψ , − β xy − (d) ψ , − β Figure 19: Some components of 3-gons associated to g β . Parts (a) and (b) show possible compo-nents of ψ β and (c) and (d) show possible components of ψ − β . ... x Figure 20: The fork diagram F .The solid arc is α n − and thedashed arc is β n − . The omittedarcs α n and β n are grayed. ... z Figure 21: The fork diagram F ′ .The solid arc is α ′ n and the dottedarc is β ′ n . The omitted arcs α n − and β n − are grayed.Now instead let z xy ∈ Z with x ∈ α n − ∩ β j and y ∈ α k ∩ β n − (where j, k < n − F containing possible choices for x and y are shown in Figure 24, andcorresponding local regions of F ′ are shown in Figure 25. Corresponding regions from the Heegaard26 (a) H , zz ′ (b) H n,n Figure 22: Local pictures of diagrams cov-ering a neighborhood of the puncture µ n − ... z Figure 23: Result of an isotopy on Fig-ure 21diagrams H , and H n,n are shown in Figures 26 and 27. One can see that g ( z x y ) ∈ (cid:8) z u v , z u ′ v , z u v ′ , z u ′ v ′ (cid:9) , g ( z x y ) ∈ (cid:8) z u v , z u ′ v (cid:9) , g ( z x y ′ ) ∈ (cid:8) z u v ′ , z u ′ v ′ (cid:9) ,g ( z x y ) ∈ (cid:8) z u v , z u ′ v , z u v ′ , z u ′ v ′ (cid:9) , g ( z x y ) ∈ (cid:8) z u v , z u ′ v (cid:9) , g ( z x y ′ ) ∈ (cid:8) z u v ′ , z u ′ v ′ (cid:9) ,g ( z x ′ y ) ∈ (cid:8) z u ′ v , z u v , z u ′ v ′ , z u v ′ (cid:9) , g ( z x ′ y ) ∈ (cid:8) z u ′ v , z u v (cid:9) , g ( z x ′ y ′ ) ∈ (cid:8) z u ′ v ′ , z u v ′ (cid:9) ......... x x (a) x i ∈ α n − ∩ β j i , i = 1 , ... ... ... ... y y (b) y i ∈ α k i ∩ β n − , i = 1 , Figure 24: Local pictures of F appearing case II of the proof of Lemma 3.9.2. One could choose x to be either x or x and y to be either y or y .Figure 28 shows the result of the isotopy making α ′ n horizontal. One can verify that by examiningFigures 24 and 28 that for 1 ≤ i ≤ T ( z u v i ) = T ( z x y i ) and T ( z u v i ) = T ( z u v i ) = T ( z x y i ).Additionally, (cid:0) P ∗ − Q ∗ (cid:1) ( u ) = (cid:0) P ∗ − Q ∗ (cid:1) ( u ) = (cid:0) P ∗ − Q ∗ (cid:1) ( x ) + 1 , (cid:0) P ∗ − Q ∗ (cid:1) ( u ) = (cid:0) P ∗ − Q ∗ (cid:1) ( x ) + 1and (cid:0) P ∗ − Q ∗ (cid:1) ( v i ) = (cid:0) P ∗ − Q ∗ (cid:1) ( y i ) + 1 for 1 ≤ i ≤ . So, g preserves R in this case as well. 27 ........ u u u (a) Elements of α n − ∩ β j i , i = 1 , ... ... ... ... v v (b) Elements of α k i ∩ β n − , i = 1 , Figure 25: Local pictures of the fork diagram F ′ x x x ′ (a) Near µ n − y (b) Near µ k − y y ′ (c) Near µ k − Figure 26: Local pictures of the Heegaard diagram H n u ′ u ′ u ′ u u u (a) Near µ n − v v ′ (b) Near µ k − v v ′ (c) Near µ k − Figure 27: Local pictures of the Heegaard diagram H ′ ( n − ......... u u u Figure 28: The result of an isotopy on Figure25(a) In light of the “cancellation rule” provided by Proposition 1.0.2, Proposition 1.0.1 indicates thatthe filtered chain homotopy type of the reduced filtered complex determines that of the unreduced28ltered complex, and vice versa - provided that we work with coefficients in F , a field. Togetherwith Theorem 1.0.1 from [Twe14], this immediately implies Theorem 1.0.3 above. The goal of thissection is to prove Proposition 1.0.2. First we recall some definitions related to filtrations andestablish some notation.Let F be a field and let V be a vector space over F equipped with a Z -filtration · · · ⊆ F i − V ⊆ F i V ⊆ F i +1 V ⊆ · · · so that S F i V = V and T F i V = ∅ . Then recall that there is an associatedgraded vector space gr( V ) defined bygr( V ) := ∞ M i = −∞ gr i ( V ) where gr i ( V ) := F i V /F i − V Given some x ∈ V , the filtration level of x is the integer f ( x ) := inf (cid:8) k ∈ Z (cid:12)(cid:12) x ∈ F k V (cid:9) . For each k ∈ Z , there is a natural map ι k : F k V → gr( V ) given by composing the projection F k V → F k V /F k − V with the inclusion F k V /F k − V → gr( V ). Then we can define a function [ · ] : V → gr( V ) via[ x ] := ι f ( x ) ( x ).Furthermore, if V = ⊕ i ∈ Z V i is a graded vector space, let g ( x ) ∈ Z denote the grading of ahomogeneous element x ∈ V . Recall that a filtered chain complex ( C, ∂ ) over a field F is a gradedvector space C over F together with a differential ∂ : C → C such that ∂ = 0 and ∂ ( F k C i ) ⊂ F k C i − for all i, k ∈ Z . Definition 3.10.1. Let C be a finite-dimensional Z -graded, Z -filtered chain complex of vectorspaces over a field F . We say that C is reduced if for every x ∈ C , f ( ∂ ( x )) < f ( x ), i.e. thedifferential strictly lowers the filtration level. Definition 3.10.2. Let V be a finite-dimensional Z -filtered vector space over a field F .(i) A set { x , . . . , x n } ⊂ V is filtered linearly independent if the set { [ x ] , . . . , [ x n ] } ⊂ gr( V ) islinearly independent in gr( V ).(ii) A basis { x , . . . , x n } for V is a filtered basis if the set { [ x ] , . . . , [ x n ] } is a basis for gr( V ).We visualize a filtered chain complex on the 1-dimensional lattice Z by choosing a filtered basis { x , . . . x n } and plotting each generator x i at the position f ( x i ) ∈ Z . The differential appears as asystem of arrows, with one arrow pointing from x i to each generator appearing in the expression ∂ ( x i ) (for each i ). We’ll refer to these arrows as the components of the differential and we’ll saythat a generator x i is incident to some component if that component either emanates or terminatesat x i . Definition 3.10.3. Let C be a finite-dimensional Z -graded, Z -filtered chain complex of vectorspaces over a field F . Then a filtered basis { x , . . . , x n } for C is simplified if each x i is incident toat most one component of the differential.We’ll need several lemmas to prove Proposition 1.0.2. The following result is proved in [LOT08]: Lemma 3.10.4 (Proposition 11.52 from [LOT08]) . Let C be a finite-dimensional Z -graded, Z -filtered chain complex of vector spaces over a field F . Then there exists a reduced filtered complex C ′ which is filtered chain homotopy equivalent to C . Furthermore, C ′ has a filtered basis that issimplified. Lemma 3.10.5. Fix a positive integer k . Let V , W , and W be finite-dimensional Z k -gradedvector spaces over a field F . If W ⊗ F V ∼ = W ⊗ F V as Z k -graded vector spaces, then in fact W ∼ = W . The above fact is of interest in its own right as a cancellation tool for graded vector spaces. Itwas presumably previously known, but the author has been unable to find a particular reference.The following lemma will allow us to prove Proposition 1.0.2: Lemma 3.10.6. Let C be a finite-dimensional Z -graded, Z -filtered chain complex of vector spacesover a field F . Then the filtered chain homotopy type of the complex C is completely determined bythe Z -graded isomorphism types of the pages { E r } r ≥ of the spectral sequence induced by C .Proof of Proposition 1.0.2. For each i and for k ≥ 1, let E i,k denote the k th page of the spectralsequence induced by the filtered complex C i . For k ≥ 1, the k th page of the spectral sequenceinduced by the complex C i ⊗ F V F is isomorphic as a Z -graded group to E i,k ⊗ F ( F (1 , ⊕ F (0 , k th page of the spectral sequence induced by C i ⊗ F W F is isomorphic to E i,k ⊗ F ( F (0 , . So,regardless of whether X = V F or X = W F , Lemma 3.10.5 implies that for each k ≥ E ,k ∼ = E ,k as Z -graded vector spaces. Now Lemma 3.10.6 implies that C and C have the same filtered chainhomotopy type. Proof of Lemma 3.10.5. For concreteness, let V = n M j =1 F ( a j ), where a j ∈ Z k . After relabeling ifnecessary, assume that m ∈ { , , . . . , n } satisfies a j = a for j ≤ m and a j = a for j > m .For i ∈ { , } , the Z k -graded isomorphism type of W i is determined by the function f i : Z k → Z ≥ , where f i ( x ) := dim F (( W i ) x ). Define an analogous function g i associated to W i ⊗ F V . Noticethat for each x ∈ Z k , g i ( x ) = n X j =1 f i ( x − a j ) and so mf i ( x ) = g i ( x + a ) − n X j = m +1 f i ( x + a − a j ) (3.3)Now Equation 3.3 implies that f ( x ) − f ( x ) only depends on the equivalence class of x in thequotient Z k /H , where H := Span { a − a j | m < j ≤ n } .Since W and W are finite-dimensional, f − f = 0 outside of a compact set in Z k . Therefore,in the case that m < n (i.e. not all of the a j are equal), f − f ≡ m = n , Equation 3.3 implies that 0 ≡ g − g ≡ n ( f − f ) and so f ≡ f since Z istorsion-free.Prior to proving Lemma 3.10.6, we’ll define some particular filtered chain complexes. Givenintegers s, t ∈ Z , let V ( s, t ) denote the Z -filtered, Z -graded chain complex of vector spaces given by V ( s, t ) := ( F h x i , d ≡ 0) where x has grading g ( x ) = s and filtration level f ( x ) = t . For integers s, t and positive integer δ , let W ( s, t, δ ) denote the Z -filtered, Z -graded chain complex of vector spacesgiven by W ( s, t, δ ) := ( F h x i ⊕ F h y i , d ) with ( g, f )( x ) = ( s, t ), ( g, f )( y ) = ( s − , t − δ ), d ( x ) = y ,and d ( y ) = 0. 30 roof of Lemma 3.10.6. By Lemma 3.10.4 above, C is filtered chain homotopy equivalent to somereduced filtered chain complex C ′ , and C ′ has a simplified basis. Then we in fact have that C ′ = n M i =1 V ( s i , t i ) ! ⊕ m M j =1 l j M k =1 W ( a j,k , b j,k , δ j ) (3.4)for some positive integers m , n , δ j and l j (for each 1 ≤ j ≤ m ), some integers s i , t i for each1 ≤ i ≤ n , and some integers a j,k , b j,k for each 1 ≤ j ≤ m and 1 ≤ k ≤ l j . It is assumed that δ < δ < . . . < δ m , and recall that δ > C ′ is reduced. Note that the summands V ( s i , t i )in Equation 3.4 are generated by elements in the simplified filtered basis for C ′ which aren’t incidentto any components of the differential, and the components of the summands W ( a j,k , b j,k , δ j ) aregenerated by elements each incident to exactly one component of the differential.Let E r denote the pages of the spectral sequence induced by the filtered complex C ′ . Firstnotice that since C ′ is reduced, E ∼ = E as Z -graded vector spaces. Furthermore, we have that E ∞ ∼ = n M i =1 F ( t i , s i − t i )Recall that for each r , E r +1 is isomorphic to a Z -graded subspace of E r , and the quotients E r +1 /E r are nonzero exactly when r = δ j for some j . Therefore, the higher pages determine the numbers δ j . Now for each j with 1 ≤ j ≤ m , E δ j +1 E δ j ∼ = l j M k =1 ( F ( b j,k , a j,k − b j,k ) ⊕ F ( b j,k − δ j , a j,k − b j,k + δ j − ∼ = l j M k =1 F ( b j,k , a j,k − b j,k ) ⊗ F ( F (0 , ⊕ F ( − δ j , δ j − δ j , then by Lemma 3.10.5, the Z -graded isomorphism type of thelast expression in Equation 3.5 determines the Z -graded isomorphism type of l j M k =1 F ( b j,k , a j,k − b j,k ) which in turn determines the numbers b j,k and a j,k for all k . So, we now know the filtered chainisomorphism type of the filtered chain complex C ′ . Recall that Ozsv´ath and Szab´o showed in [OS04a] that the relatively-graded Heegaard Floer chaincomplexes satisfy a K¨unneth-type relationship under connected sums of 3-manifolds. We state theresult for d CF : 31 heorem 4.1.1 (Proposition 6.1 from [OS04a]) . Let M and M be oriented -manifolds withSpin c structures s i ∈ Spin c ( M i ) for i = 1 , . Then d CF ( M M , s s ) ∼ = d CF ( M , s ) ⊗ Z d CF ( M , s ) . Recall also that for M a rational homology 3-sphere with s ∈ Spin c ( M ), Ozsv´ath and Szab´odefined in [OS03a] the correction term d ( M, s ) ∈ Q to be the minimal absolute grading e gr of anynon-torsion element in the image of HF ∞ ( M, s ) in HF + ( M, s ). It was shown in [OS03a] that d ( M M , s s ) = d ( M , s ) + d ( M , s ) . It follows that when M and M are rational homology 3-spheres, the d CF complexes satisfy aK¨unneth formula as absolutely-graded chain complexes with grading e gr , i.e. d CF e gr = k ( M M , s s ) ∼ = M i + j = k (cid:18)d CF e gr = i ( M , s ) ⊗ Z d CF e gr = j ( M , s ) (cid:19) . (4.1)As stated in Theorem 1.0.7, when M i = Σ( K i ) for i = 1 , 2, then the complexes for M , M , andΣ( K K ) ∼ = M M satisfy a K¨unneth-type relationship as filtered complexes up to filtered chainhomotopy type. Proof of Theorem 1.0.7. Let K and K be knots in S . Then we can find braids b ∈ B m − and b ∈ B n − such that the plat closure of b ′ = (cid:0) b × σ − × (cid:1) ∈ B m +2 is a diagram D for K and the plat closure of b ′ = (1 × b × ∈ B n is a diagram D for K . Then the plat closureof b = (cid:0) b × σ − × b × (cid:1) ∈ B n +2 m is a diagram D for K = K K (where we view σ ∈ B ).These plat closures are illustrated in Figure 29. ...... b (a) D ...... b (b) D ............ b b (c) D Figure 29: Plat closures of b ′ ∈ B m +2 , b ′ ∈ B n , and b ∈ B m +2 n ... ... y (a) The fork diagram F with the arcs α m +1 , β m +1 grayed ... ... z z (b) The fork diagram F with the arcs α n , β n grayed Figure 30: Local pictures of fork diagrams for summand knots K i .. ...... ... y v v Figure 31: The fork diagram F for K K with the arcs α m + n , β m + n grayed.Let F , F , and F denote the reducible fork diagrams induced by b ′ , b ′ , and b , respectively.We choose the braids b i in such a way that the fork diagrams F i have the local behavior indicatedin Figure 30; this can always be achieved through stabilization. The fork diagram F can be seenin Figure 31.We’ll compute the reduced gradings R by omitting the pair α m , β m from F , the pair α n , β n from F , and the pair α m + n , β m + n from F .Figure 32 shows the Heegaard diagrams H , H , and H covering F , F , and F , respectively.Let the sets of attaching circles for these diagrams be denoted by { α , β } , { α , β } , and { α , β } ,respectively. ... ... yaa bb (a) H ... ... z z cc dd (b) H ... ... ... ... y v v ′ v aa bb cc dd (c) H Figure 32: Heegaard diagrams depicting local connected sum behaviorRecall that if the connected sum region in each Heegaard diagram is near the basepoint, then H , H , and H = H H are Heegaard diagrams exhibiting the correspondences in Theorem4.1.1 and Equation 4.1. The sets of attaching circles for H are { α , β } = { α ⊔ α , β ⊔ β } .The set α can be obtained from the set α via a sequence of m pointed handleslides α = e α e α . . . e α m = α , ..... a b c d ea b c d e (a) α = e α e α ...... a b c d ea b c d e (b) e α e α ...... a b c d ea b c d e (c) e α e α = α Figure 33: Handleslides connecting the sets α and α . Notice that the first two handleslides areof the form shown in Figure 2. In each picture, the old curve is dashed, the new curve is dotted,unaffected curves are solid, and the pair of pants is shaded.as depicted in Figure 33 for m = 3. The set β can be obtained from the set β by an analogoussequence of m pointed handleslides. We then study the sequence of pointed Heegaard diagrams H , 7→ H , . . . 7→ H m − , 7→ H m − , . . . 7→ H m − ,m − 7→ H m,m − 7→ H m,m . (4.2)where H i,j = (cid:16) Σ; e α i ; e β j ; + ∞ (cid:17) . Now H , = H and H m,m = H are automatically admissiblebecause they cover fork diagrams. The first 2 m − H m,m − and H m,m , indeed H m,m − can alternately beobtained from H m,m by a sequence of handleslides and isotopies like those in Lemmas 2.0.22 and2.0.21. So, all intermediate diagrams in the sequence in Equation 4.2 are admissible.For the first 2 m − e g : T e α ∩ T e β → T e α m − ∩ T e β m − their composition.As in Section 3.9.1, we define triangle injections g α : T e α m ∩ T e β m − → T e α m − ∩ T e β m − and g β : T e α m ∩ T e β m → T e α m ∩ T e β m − using the 3-gons ψ ± α and ψ ± β for various generators, whose components are found in Figures 34 and35 (components not shown are like those in Figure 16).One can see that Im ( e g ) ⊆ Im ( g α ◦ g β ), and so we can define g = ( g α ◦ g β ) − ◦ e g : T α ∩ T β → T α ∩ T β . Let x y be a generator in H and let w z be a generator in H (where x is a ( m − w isa ( n − y ∈ α m , and z ∈ α ). The generators in H are exactly of the form xw yz , and weset R ( xw yz ) = R ( x y ) + R ( w z ). It isn’t hard to see that g ( xw yz i ) = xw yv i for i = 1 , . a) ψ + α v (b) ψ + α (c) ψ + α (d) ψ + α (e) ψ − α v (f) ψ − α (g) ψ − α (h) ψ − α Figure 34: Some components of domains for the various 3-gons associated to a α -triangle injection.In each figure, the e α m are dashed, the e α m − are dotted, the e β m − are solid, the dark grey dot isa component of some x , and the light grey dot is the corresponding component of g α ( x ). (a) ψ + β v ′ (b) ψ + β v (c) ψ + β (d) ψ − β v ′ (e) ψ − β v (f) ψ − β Figure 35: Some components of domains for the various 3-gons associated to a β -triangle injection.In each figure, the e β m are dashed, the e β m − are dotted, the e α m are solid, the dark grey dot is acomponent of some x , and the light grey dot is the corresponding component of g β ( x ).By examining the fork diagrams in Figures 30 and 31, one can verify that indeed e R ( xw yv i ) = e R ( x y ) + e R ( w z i ) for i = 1 , 2. Further, notice that e ( b ) = e ( b ′ ) + e ( b ′ ) and w ( D ) = w ( D ) + w ( D ).Therefore, s R ( b, D ) = e ( b ) − w ( D ) − m + n − e ( b ′ ) − w ( D ) − m − e ( b ′ ) − w ( D ) − n − s R ( b , D ) + s R ( b , D ) . .1.2 Knot mirrors Given a braid word b = σ k i . . . σ k m i m ∈ B n , let − b denote the braid word − b = σ − k n − i . . . σ − k m n − i m ∈ B n . Notice that if the plat closure of b is K , then the plat closure of − b is − K , the mirror of K . Proof of Theorem 1.0.9. Let b ∈ B n be a braid whose closure is the knot K , and let H ± denotethe admissible Heegaard diagram for Σ( ± K ) induced by ± b . Let s ∈ Spin c (Σ( K )). Then by anargument analogous to that in the proof of Lemma 33 in [Twe14], the natural isomorphismΦ : d CF ∗ (cid:0) H + , s (cid:1) → d CF −∗ (cid:0) H − , s (cid:1) is filtered when one equips the complexes with the filtrations R and R ∗ , respectively. In addition,if one equips d CF ( H − , s ) with an absolute grading g on its generators given by g ( x ∗ ) = − e gr ( x ),then Φ is graded with respect to e gr and g . So, Φ and Φ − are filtered with respect to the filtrations ρ and ρ ∗ . Q -valued knot invariants The filtration R gives rise to another more concise knot invariant. Let K be a knot which is theclosure of a braid b , let H be the Heegaard diagram for Σ( K ) induced by b , and let s ∈ Spin c (Σ( K )).Let F ( b, r, s ) ⊂ d CF ( H , s ) denote the subcomplex generated by intersection points whose filtrationlevel is at most r ∈ Q . For each s ∈ Spin c (Σ( K )) and each r ∈ Q , one obtains the homomorphisminduced by inclusion: ι rK, s : H ∗ ( F ( b, r, s ) , b ∂ ) → H ∗ ( d CF ( H , s ) , b ∂ ) = d HF (Σ( K ) , s ) . Then define functions r minK , r maxK , r K : Spin c (Σ( K )) → Q given by r minK ( s ) = min { r ∈ Q | ι rK, s = 0 } ,r maxK ( s ) = min { r ∈ Q | ι rK, s is an isomorphism } , and r K ( s ) = 12 (cid:0) r maxK ( s ) + r minK ( s ) (cid:1) . The maps ι rK, s don’t depend on the choice of b , and so the above functions are indeed knotinvariants.We then set r ( K ) = r K ( s ), where s ∈ Spin c (Σ( K )) is the unique Spin structure. The readershould compare this definition to those of τ from [OS03b], s from [Ras10], and δ from [MO07].In [MO07], Manolescu and Owens defined a concordance invariant via δ ( K ) = 2 d (Σ( K ) , s ),where d denotes the correction term defined by Ozsv´ath and Szab´o in [OS03a] and s ∈ Spin c (Σ( K ))is the unique Spin structure. Furthermore, it was shown in [MO07] that whenever K is an alter-nating knot, δ ( K ) = − σ ( K ) / . Together with Theorem 1.0.14 above, this implies that if K is atwo-bridge knot, then r ( K ) = σ ( K )2 − d (Σ( K ) , s ) = σ ( K )2 − δ ( K )2 = 3 σ ( K )4 . .3 Degeneracy The unreduced filtration ρ described in [Twe14] induces a spectral sequence converging to the group d HF (Σ( K ) S × S )). Here we’ll state a version of a definition from [Twe14], modified to the caseof coefficients in a field F : Definition 4.3.1. Let F be a field. A knot K is called ρ -degenerate if the following hold:(i) The spectral sequence induced by ρ (wth coefficients in F ) converges at the E -page.(ii) The filtration on d HF (Σ( K ) S × S ); F ) induced by ρ is constant on each Spin c -summand d HF (Σ( K ) S × S ) , s ; F ).Proposition 1.0.1 implies that a knot is ρ -degenerate if and only if the reduced spectral sequenceover F (induced by ρ ) satisfies properties analogous to those listed in Definition 4.3.1. Since R − ρ = e gr , Proposition 1.0.13 follows.Notice that even when a knot is ρ -degenerate, the value taken by r ( K ) may still be of interest. Assuggested in Section 6.2, one could ask whether r itself provides a concordance invariant and couldcompare it other known concordance invariants arising in Heegaard Floer theory and Khovanovhomology. Furthermore, it would be interesting to investigate whether r = 3 σ/ L (3 , Let K be the left-handed trefoil, viewed as the plat closure of the braid σ ∈ B . This braid wasstudied in Section 3.2 of [Twe14], and the fork diagram shown there is reducible. We can omitthe pair α , β from that fork diagram and obtain via Proposition 3.5.1 an admissible Heegaarddiagram for L (3 , 1) of genus 1; this diagram can be seen in Figure 36. Let the generators be labelledfrom left to right as t ′ , t, and x + ∞ b α b β a a Figure 36: Heegaard diagram for L (3 , 1) ob-tained from σ ∈ B The set of reduced Bigelow generators is G = { t, t ′ , x } , and one can verify that all ele-ments occupy the level R = 1 . In this case b ∂ ≡ 0, and indeed we see evidence of ρ -degeneracy(which was already found in [Twe14]). Thus R provides an absolute Maslov grading on the group d HF ( L (3 , Z / Z ), and d HF ( L (3 , Z / Z ) = h ( Z / Z ) ⊕ i R =1 . One should observe that d HF ( L (3 , Z / Z ) is supported entirely in the grading level R = 1(and 1 is half of the classical signature σ ( K )). Theorem 1.0.14 states that all two-bridge knotsbehave this way. 37 .5 Two-bridge knots Let us first give some background on two-bridge knots and links. Recall that a two-bridge knot is a knot which has a projection on which the natural height functionhas exactly two maxima and two minima. A two-bridge link is defined similarly, with exactly onemaximum and one minimum of the height function lying on each of the two components. ...... ...... ...... b b k b (a) k odd ...... ... ......... ...... ... ......... b b k b b b k -1 b (b) k even Figure 37: Conway forms for knots orlinks with Conway notation [ b , . . . , b k ]with all b j > 0. The label “ b j ” indi-cates a bundle of b j half-twists of theindicated direction.For each two-bridge knot or link L , there are nonzero integers b , b , . . . , b k all of the same signsuch that one of the two diagrams in Figure 37 is a projection of either L or its mirror image (themirror image must be taken when the numbers b j are negative). After possibly performing theisotopy shown in Figure 37(b), we can assume that k is odd.This diagram is referred to as the Conway form, with Conway notation given by the continuedfraction [ b , . . . , b k ] = b + 1 b + 1 b + 1 b + . . . = pq . Remark . If L is a knot or link with Conway notation [ b , . . . , b k ] = pq , p and q coprime, thenthe two-fold cover of S branched along L is the lens space L ( p, q ).For more about the Conway form and Conway notation, see [Rol03]. Gordon and Litherland give a formula in [GL78] for calculating the signature σ ( K ) of a knot K .Here we review their construction briefly, but one can find more details in [GL78].Given a regular projection D of a knot K in R , color the components of R — D black and whitein a checkerboard fashion, denoting the white regions by X , . . . , X n . Denote by c ( X i , X j ) the setof crossings of D which are incident to X i and X j . Then assign an incidence number η ( C ) = ± C in the projection, following the convention in Figure 38.38 ( C ) = +1 η ( C ) = − Figure 38: Conventions for η ( C ), where C isa crossing in a colored diagram D Definition 4.5.2. Let D be a regular projection of a knot K equipped with a checkerboard coloringand white regions X , . . . , X n . Then let G ′ ( D ) be the ( n + 1) × ( n + 1) matrix with G ′ ( D ) = ( g ij ) ni,j =0 , where g ij = − X c ( X i ,X j ) η ( C ) if i = j − X k = i g ik if i = j. Then define G ( D ), the Goeritz matrix of D , to be the n × n symmetric integer matrix obtainedfrom G ′ ( D ) by deleting the 0 th row and 0 th column. Type I Type II Figure 39: Types of double point orienta-tions counted by µ I and µ II Now fixing an orientation on K , we separate the double points of D into types I and II, accordingto Figure 39. Note that when classifying a crossing in this way, we ignore which strand is passingover the other.Now we define two integers µ I and µ II by µ I ( D ) = X C of type I η ( C ) and µ II ( D ) = X C of type II η ( C ) . It is shown in [GL78] that the quantity sign ( G ( D )) − µ II ( D ) is independent of the projection D , and thus an invariant of the knot K . We’ll make use of the following theorem: Theorem 4.5.3 (Theorem 6 from [GL78]) . Let K be a knot with regular projection D . Then σ ( K ) = sign ( G ( D )) − µ II ( D ) . We’ll need the following fact, the proof of which will be left as an exercise. Lemma 4.5.4. Let K be a knot with oriented regular projection D . Then w ( D ) = µ II ( D ) − µ I ( D ) . .5.3 Computations for two-bridge knots We’ll perform our calculation for a two-bridge knot K using the braid whose closure is the Conwayform of K . Further, we assume that the number of Bigelow generators cannot be reduced by anisotopy of the fork diagram. Because Conway forms never involve the rightmost strand in thebraiding, the induced fork diagram will always be reducible. After reducing the diagram, G will bethe set of 1-tuples α ∩ bE ′ . First we show that the function R is very simple for such a reducedfork diagram: Proposition 4.5.5. Let K be a two-bridge knot with Conway notation [ b , . . . , b k ] . Then for anyreduced Bigelow generator x ∈ G in the special reduced fork diagram above, R ( x ) = ( e − w − if b > e − w +24 if b < . where e is the signed count of braid generators in the Conway form and w is the writhe of thatdiagram. Before proving Proposition 4.5.5, we’ll have to define some new terminology. Definition 4.5.6. Let K be an oriented two-bridge knot, and consider the special fork diagramacquired from the Conway form of K . Augment the diagram by adding an extra horizontal tineedge α connecting µ and µ , and give it a vertical handle h . Then we’ll call x ∈ ( α − µ ) ∩ β a central intersection .We then extend the e R grading to central intersections in the natural way. Of course our reducedfork diagrams don’t include α or its handle; we will simply use these central intersections as aninductive tool for proving Proposition 4.5.5. Proof of Proposition 4.5.5. We first prove the proposition for the case where b i > i = 1 , . . . , k .The fork diagram corresponding to only σ b includes b Bigelow generators with e R = 0. Noticethat in such a diagram, when β is incident to µ i , it approaches from below when i = 1 or i = 3and from above when i = 2. Furthermore, the actions of σ and σ − don’t affect the directionsfrom which β approaches these punctures. Let us develop inductive steps for applying σ − and σ , the building blocks for diagrams of this type.For the first inductive step, we examine the application of σ − to an existing braid b . Eachexisting element g ∈ G on the interior of α spawns one new central intersection c and is itselfreplaced by another interior g ′ ∈ G , with e R ( g ′ ) = e R ( g ) and e R ( c ) = e R ( g ) + 1.We should also examine the effects on an arc terminating at µ (case I) or µ (case II), as inFigure 40.In both cases, the element g ∈ G is replaced by g ′ ∈ G . We see that e R ( g ′ ) = e R ( g ).In case II, the application of σ − also spawns a new interior central intersection point, c . Wesee that e R ( c ) = e R ( g ) + 1 . The loops used for grading calculations above can be seen in Figure 41.For the second inductive step, we examine the application of σ to an existing braid b . Eachexisting central intersection c spawns a new element g ∈ G on the interior of α , and is itselfreplaced by a new central intersection c ′ . We see that e R ( c ′ ) = e R ( c ) and e R ( g ) = e R ( c ) − a) Case I (b) Case II Figure 40: The endpoint cases in the σ − inductive step. The arcs α and α are dotted black anddotted gray, respectively. g g ′ (a) Case I g g ′ c (b) Case II Figure 41: Grading loops associated to the σ − inductive stepSome new elements of e G can also result from twisting a strand that originally terminated at µ (case I) or µ (case II), as shown in Figure 42.In case I, we gain a central intersection c , and the element g ∈ G is replaced by h ∈ G . In caseII, the central intersection c is replaced by g ∈ G . In either case, we have that e R ( c ) = e R ( g ) + 1 and e R ( h ) = e R ( g ).The grading comparisons calculated above are demonstrated by the loops in Figure 43.Using the results of these two inductive steps, we see that all elements of e G for the b > e R grading.The proof for the b i < σ and σ − . The fork diagram corresponding to only σ b includes − b Bigelowgenerators with e R = 1. So, all elements of e G lie in level 1 of the e R grading when b < x ∈ G , e R ( x ) = ( b > 01 if b < . and the result for R follows. Lemma 4.5.7. Let k be odd and let b , b , . . . , b k be nonzero integers which are all of the samesign. If the knot projection D is given by the plat closure of σ b σ − b . . . σ − b k − σ b k (as in Figure37(a) or its mirror image), then sign ( G ( D )) = − (cid:18) X i even b i (cid:19) − if b > − (cid:18) X i even b i (cid:19) + 1 if b < . When proving Lemma 4.5.7, we’ll make use of the following fact.41 a) Case I (b) Case II Figure 42: The endpoint cases in the σ inductive step g c h (a) Case I c g (b) Case II Figure 43: Grading loops associated to the σ inductive step Lemma 4.5.8. For i = 1 , , . . . , m , choose real numbers a i ≥ . Then the matrix given by A m := a − − a . . . − . . . − . . . a m − − − a m is positive-definite (the entries specified are the only non-zero ones).Proof of Lemma 4.5.8. For any X := ( x , x , . . . , x m ) T ∈ R m , X T A m X = x ( a x − x ) + x m ( a m x m − x m − ) + m − X i =2 x i ( a i x i − x i − − x i +1 ) ≥ x (2 x − x ) + x m (2 x m − x m − ) + m − X i =2 x i (2 x i − x i − − x i +1 )= x + x m + ( n − X i =1 ( x i − x i +1 ) . Proof of Lemma 4.5.7. We first consider the case in which b > 0. Color the components of R — K such that the exterior region is black. Then there are b + 1 white regions, where b := X i even b i + 1.Label the white regions such that X is the region on the right side of the Conway form diagramand X , . . . , X b are the ones on the left, labelled from top to bottom. The Goeritz matrix G ( D ) is42 ( b × b ) matrix given by a a . . .1 . . . 1. . . a b − a b where a i = − ( b j + 1) if i = b or i = b k − ( b j + 2) if i = b j with 1 ≤ j ≤ k − G ( D ) is negative-definite by Lemma 4.5.8, and sign ( G ( D )) = − b = − (cid:18) X i even b i (cid:19) − . When b < 0, the Goeritz matrix is positive-definite and has signature sign ( G ( D )) = − (cid:18) X i even b i (cid:19) + 1 . Proof of Theorem 1.0.14. By Proposition 4.5.5, all generators have the same R -filtration value.However, Equation 3.6 implies that nonzero components of the differential b ∂ on the filtered complexmust decrease the R -level by at least one; consequently, b ∂ ≡ 0. The knot K is thus ρ -degenerate,since rk (cid:16) d HF (Σ( K ) , s ) (cid:17) = 1 for each s ∈ Spin c (Σ( K )). For b > 0, we have that e − w − 24 = 14 (cid:18) k X i =1 ( − i +1 b i (cid:19) + µ I − µ II − ! , and σ ( K ) = sign ( G ( D )) − µ II ( D ) = − X i even b i ! − − µ II ( D ) . Furthermore, it is easy to see that − ( µ I ( D ) + µ II ( D )) = k X i =1 b i . So, for each generator x , we then have that2 R ( x ) − σ ( K ) = 12 (cid:18) k X i =1 ( − i +1 b i (cid:19) + 2 (cid:18) X i even b i (cid:19) + µ I + µ II ! = 12 (cid:18) k X i =1 b i (cid:19) + µ I + µ II ! = 12 (0) = 0 . The calculation for the b < LINKS As mentioned in Remark 3.8.3, the constructions appearing here (and indeed, the unreduced con-structions appearing in [Twe14]) can be partially extended to the case of a link of several compo-nents. The proof of invariance of filtered chain homotopy type in the unreduced case (Theorem 1of [Twe14]) doesn’t rely on the number of components, and so carries through to the case of a link.A version of Proposition 1.0.1 holds for links, and so an analogue of Theorem 1.0.3 does as well (aslong as one restricts to torsion Spin c -structures). In particular, although it can be arranged (bychoosing the right braid representative) that the deleted pair α n , β n belongs to any link componentwe like, the ρ -filtered chain homotopy type of d CF doesn’t depend on this choice.If L is a two-bridge link, then b (Σ( L )) = 0, and so all Spin c -structures are torsion. This allowsone to define the filtrations R and ρ on the entire complex d CF (Σ( L )), and indeed Theorem 1.0.14holds for links as well (with the proof unchanged). ρ -degeneracy We have seen that when the knot K is ρ -degenerate, then the function R provides an absoluteMaslov grading on the group d HF (Σ( K )). It was established above that this occurs when K is a two-bridge knot. It is natural to ask whether a larger class of knots is ρ -degenerate, suchas alternating knots or quasi-alternating knots. Following Seidel and Smith in [SS10], we haveconjectured above that all knots are ρ -degenerate. Knot homology theories and Floer homology theories have been known to produce invariants of theknot concordance class. Examples include Rasmussen’s s -invariant in Khovanov homology [Ras10],Manolescu and Owens’s δ -invariant in Heegaard Floer homology [MO07], and Ozsv´ath and Szab´o’s τ -invariant in Knot Floer homology [OS03b]. Conjecture 1.0.12 speculates that r also provides one.Recall that the concordance invariants s , δ , and τ have all been shown to be equal to a constantmultiple of σ (the classical signature) when restricted to the set of alternating knots. We saw inSection 4.2 that r = 3 σ/ r to s , δ , and τ . Bloom showed in [Blo10] that odd Khovanov homology is invariant under Conway mutation, whileOzsv´ath and Szab´o proved in [OS04c] that knot Floer homology can distinguish a Kinoshita-Terasaka knot from one of its mutants. Viro noted in [Vir76] that mutant links have homeomorphicdouble branched covers, and thus can’t be distinguished by the Heegaard Floer homology groupsdiscussed here. However, one could ask whether this extra filtration structure on the chain complexcan distinguish mutant knots. Whether or not the mutants are ρ -degenerate, it is possible that theinvariant r could be used to distinguish them. 44 cknowledgements It is my pleasure to thank Ciprian Manolescu for suggesting this problem to me and for his in-valuable guidance as an advisor. I would also like to thank Robert Lipshitz, Liam Watson, andTye Lidman for some instructive discussions, Stephen Bigelow for some helpful email related to hispaper [Big02], Yi Ni for some useful comments regarding relative Maslov gradings, and SucharitSarkar for suggesting a definition for r K . Many thanks to Shelly Harvey for suggesting the proofof Lemma 3.10.5 given here.I am also greatly indebted to the anonymous Referee, whose corrections and suggestions led tocountless improvements to this article. References [Big02] S. Bigelow. A homological definition of the Jones polynomial. In Invariants of knots and3-manifolds (Kyoto, 2001) , volume 4 of Geom. Topol. Monogr. , pages 29–41 (electronic).Geom. Topol. 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Funct.Anal. , 20(6):1464–1501, 2010.[SW10] S. Sarkar and J. Wang. An algorithm for computing some Heegaard Floer homologies. Annals of Math. , 171:1213–1236, 2010.[Twe14] E. Tweedy. On the anti-diagonal filtration for the Heegaard Floer chain complex of abranched double-cover. J. Symplectic Geom. , 12(2):313–363, 2014.[Vir76] O. Ja. Viro. Nonprojecting isotopies and knots with homeomorphic coverings.