Transverse braids and combinatorial knot Floer homology
TTransverse braids and combinatorial knot Floer homology
Peter Lambert-Cole and David Shea Vela-Vick A bstract . We describe a new method for combinatorially computing the transverse invariantin knot Floer homology. Previous work of the authors and Stone used braid diagrams tocombinatorially compute knot Floer homology of braid closures. However, that approach wasunable to explicitly identify the invariant of transverse links that naturally appears in braiddiagrams. In this paper, we improve the previous approach in order to compute the transverseinvariant. We define a new combinatorial complex that computes knot Floer homology andidentify the BRAID invariant of transverse knots and links in the homology of this complex. . Introduction Every link L transverse to the standard contact structure on S is the closure of braid ρ that is unique up to conjugation and positive stabilization [ Ben , Wri ]. For any braid ρ ∈ B n , there is a natural multi-pointed Heegaard diagram H ρ associated to the corre-sponding braid closure. This braid diagram is a classic Heegaard decomposition of thelink complement, modified to encode the braiding. The diagram H ρ determines a bigradedcomplex (cid:91) CFK ( H ρ ) whose homology is the knot Floer homology of the mirror of the braidclosure. In addition, the diagram determines a closed generator x [ H ρ ] whose homologyclass in (cid:91) HFK ( H ρ ) is an invariant of the transverse link determined by ρ . This class t ( L ) is the BRAID invariant of the transverse link L , introduced by Baldwin, V´ertesi, and thesecond author [ BVV ].BRAID is equivalent to two other powerful transverse link invariants arising in knotFloer homology, GRID and LOSS [ BVV ]. Ozsv´ath, Szab ´o, and Thurston introduced theGRID invariant, which takes values in the grid version of knot Floer homology [ OST ] andis easily computed for knots with small grid number. Lisca, Ozsv´ath, Stipsicz, and Szab ´othen used open book decompositions to define an invariant, commonly referred to as LOSS,of (null-homologous) Legendrian and transverse knots in arbitrary –manifolds [ LOSS ].These invariants have been successfully applied to distinguish and classify transverse rep-resentatives of various knot types. Mathematics Subject Classification. M ; R , R . Key words and phrases.
Heegaard Floer homology, contact structures, transverse knots. a r X i v : . [ m a t h . S G ] M a r P. LAMBERT-COLE AND D.S. VELA-VICK
The BRAID invariant possesses advantages over its predecessors. Most notably, it is amanifestly transverse invariant as it is defined in terms of transverse knots and links that arebraided about open books. When the grid size, or equivalently arc index, exceeds the highteens, GRID cannot be efficiently computed. As the braid complexity of a transverse linkis relatively independent of arc index, BRAID potentially expands our ability to effectivelydistinguish transverse knots.Together with Stone [
LSV ], the authors used this construction to give a new, combina-torial method for computing knot Floer homology for knots and links in S . By a sequenceof stabilizations and isotopies, the diagram H ρ is modified to H nice . This diagram is nice inthe sense of Sarkar and Wang [ SW ] and thus the differential on the complex (cid:103) CFK ( H nice ) can be computed explicitly. Pseudo-holomorphic curve techniques give a chain homotopyequivalence F : (cid:103) CFK ( H ρ ) → (cid:103) CFK ( H nice ) . Thus, the bigraded ranks of (cid:93) HFK of the braid clo-sure can be computed combinatorially. However, the chain map F itself, and in particularthe image of the BRAID invariant, could not be computed explicitly.In this paper, we describe an algebraic method to compute BRAID and prove the fol-lowing theorem.T heorem 1 . . Let ρ ∈ B n be a braid with transverse link closure L. There is an associatedcomplex C ( H ρ ) that is combinatorially computable and an isomorphism on homology ( GF ) ∗ : (cid:93) HFK ( H ρ ) → H ∗ ( C ( H ρ )) This complex supports a canonical generator x [ H ρ ] such that ( GF ) ∗ ( t ( L )) = (cid:2) x [ H ρ ] (cid:3) Specifically, we show how to identify the transverse invariant from the complex (cid:103)
CFK ( H nice ) without computing the chain map F . Starting with the pair of multi-pointed Heegaard di-agrams H ρ and H nice , we define a new chain complex C ( H ρ ) . This new chain complex ishomotopy equivalent to (cid:103) CFK ( H ρ ) and possesses the same underlying module as (cid:103) CFK ( H ρ ) .The differential is defined analogously to that of (cid:103) CFK ( H ρ ) by assigning counts to Whitneydisks via the rule d ( x ) : = ∑ y ∈ T α ∩ T β ∑ φ ∈ π ( x , y ) µ ( φ )= n z ( φ )= n w ( φ )= N ( φ ) · y .However, the count N ( φ ) of representatives of a domain φ is determined combinatoriallyby the differential on (cid:103) CFK ( H nice ) , instead of geometrically by counting pseudo-holomorphicrepresentatives. The differential on (cid:103) CFK ( H nice ) determines a chain-homotopy equivalence G : (cid:103) CFK ( H nice ) → C ( H ρ ) as well. The resulting composition GF can similarly be interpretedas a standard triangle map defined by assigning modified counts to Whitney triangles.Importantly, while we cannot identify the image of x [ H ρ ] under F itself, in Proposition . we are able to determine its image under the composition GF , showing GF ( x [ H ρ ]) = x [ H ρ ] ( ) RANSVERSE BRAIDS AND COMBINATORIAL KNOT FLOER HOMOLOGY As a consequence, the transverse invariant t ( L ) is combinatorially computable directly fromthe braid. Acknowledgements.
We would like to acknowledge the National Science Foundationand Louisiana State University for sponsoring the
LSU Research Experience for Un-dergraduates (NSF Grant DMS- ). The mathematics presented here originated as anoffshoot of this program. Vela-Vick would also like to acknowledge partial support fromNSF Grant DMS- . . Preliminaries In what follows, we assume familiarity with knot and braid theory, as well as elementaryaspects of contact geometry and Legendrian and transverse links. The interested reader isencouraged to consult Birman’s book [
Bir ] and Etnyre’s notes [ Etn ] for comprehensiveintroductions to braid theory and to Legendrian and transverse knot theory. . . Knot Floer homology. We begin by summarizing some basic definitions and resultsconcerning knot Floer homology. We refer the reader to the papers by Ozsv´ath and Szab ´o[ OS b ] and Rasmusen [ Ras ] for a more in-depth discussion of this material. Throughoutthis manuscript, we work with F = Z /2 Z –coefficients.Recall that to each oriented link K in the –sphere, one can associate a multi-pointedHeegaard diagram H = ( Σ , α , β , z , w ) . In this case, the triple ( Σ , α , β ) specifies a Heegaarddiagram for S and the link K is obtained from the basepoints z = { z i } and w = { w i } byconnecting the z to w -basepoints and w to z -basepoints by properly embedded arcs in the α and β -handlebodies respectively which avoids the compression disks specified by the curvesin α and β . More generally, a multi-pointed Heegaard triple H = ( Σ , α , β , γ , z , w ) is a collectionof three sets of curves α , β , γ such that each pair ( α , β ) , ( α , γ ) , and ( β , γ ) determines multi-pointed Heegaard diagrams.The collections α and β specify tori T α and T β in Sym g + n − ( Σ ) . The complex (cid:103) CFK ( H ) is the F -vector space freely generated by the intersections in T α ∩ T β . For each Whitneydisk φ ∈ π ( x , y ) , we let n z i ( φ ) and n w i ( φ ) denote the local multiplicity of φ at z i and w i respectively. We denote by n z ( φ ) and n w ( φ ) the sums of the local multiplicities at all ofthe z and w -basepoints respectively. The chain group can be endowed with two absolutegradings, the Maslov (homological) grading M ( x ) and Alexander grading A ( x ) , which aredetermined up to an overall shift by the formulas M ( x ) − M ( y ) = µ ( φ ) − n w ( φ ) and A ( x ) − A ( y ) = n z ( φ ) − n w ( φ ) ,where φ ∈ π ( x , y ) and µ ( φ ) denote the Maslov index of the Whitney disk φ . The differen-tial on the complex (cid:103) CFK ( H ) is defined by (cid:101) ∂ ( x ) = ∑ y ∈ T α ∩ T β ∑ φ ∈ π ( x , y ) , µ ( φ )= n z ( φ )= n w ( φ )= (cid:99) M ( φ ) · y . P. LAMBERT-COLE AND D.S. VELA-VICK
The tilde version of knot Floer homology is then (cid:93)
HFK ( K ) : = H ∗ ( (cid:103) CFK ( H ) , (cid:101) ∂ ) ,and is an invariant of the link K and the number of z or w -basepoints. Its relation to the hat version of knot Floer homology is given by (cid:93) HFK ( K ) = (cid:91) HFK ( K ) ⊗ V ⊗ n ,where V is a –dimensional vector space supported in bi-gradings (
0, 0 ) and ( − − ) .Let Π α , β denote the group of periodic domains in the Heegaard diagram H = ( Σ , α , β , z , w ) .Recall that a -chain is periodic if its boundary is the union of some number of α and β curves. Let Π α , β denote the subgroup of periodic domains that avoid w ∪ z . As a group, Π α , β is isomorphic to H ( S \ K ; Z ) . The Heegaard diagram H is admissible if every domainin Π α , β has both positive and negative multiplicities. The groups Π α , β , γ , Π α , β , γ of periodicdomains and admissibility are defined similarly for a triple α , β , γ .Finally, given an admissible Heegaard triple H = ( Σ , α , β , γ , z , w ) , there is an inducedchain map F : (cid:103) CFK ( H α , β ) ⊗ (cid:103) CFK ( H β , γ ) → (cid:103) CFK ( H α , γ ) defined as F ( x ⊗ x ) : = ∑ y ∈ T α ∩ T β ∑ ψ ∈ π ( x , x , y ) µ ( ψ )= n z ( ψ )= n w ( ψ )= (cid:99) M ( ψ ) · y . . Combinatorial computations. In [
LSV ], Stone and the authors described an al-gorithm for combinatorially computing knot Floer homology. The algorithm begins with abraid presentation of a given link K and produces an explicit, nice multi-pointed Heegaarddiagram in the sense of Sarkar and Wang [ SW ]. We outline below the construction from[ LSV ].A multi-pointed Heegaard diagram H is nice if every region in Σ \ ( α ∪ β ) which doesnot containing a basepoint is topologically a disk with at most four corners. In other words,every region in the complement of the α and β -curves either contains a z -basepoint, oris a bigon or square. If a multi-pointed Heegaard diagram H is nice, the differential on (cid:103) CFK ( H ) can be computed combinatorially by counting embedded, empty rectangles andbigons connecting generators [ SW ].There is a well-known isomorphism between the n -stranded braid group B n and themapping class group MCG ( D , n ) of the disk with n marked points. Let ( D , n ) denote theunit disk in R with n evenly spaced marked points z = { z , . . . , z n } along the horizontalaxis. The isomorphism identifies the i th standard generator σ i of the Artin braid group withthe positive half-twist about the horizontal arc joining the i th and ( i + ) st marked points on D . Let a = { a , . . . , a n − } denote an arc-basis for ( D , z ) consisting of n − D \ a contains exactly one of the z -basepoints. Next, let b denotea second arc-basis which is obtained from a by applying small isotopies which shift the RANSVERSE BRAIDS AND COMBINATORIAL KNOT FLOER HOMOLOGY γ γ F igure 1 . A Dehn half-twist τ γ about the arc γ .endpoints of the a i along the orientation of ∂ D and results in a single transverse intersection x i = a i ∩ b i .Take a second copy D (cid:48) of the disk D with identical basepoints w = { w , . . . , w n } . Fora homeomorphism ρ : ( D , z ) → ( D , z ) , we endow D (cid:48) with two arc bases: the first a (cid:48) isidentical to a , while the second b (cid:48) is obtained from b by setting b (cid:48) i : = ρ ( b i ) . By perturbing,if necessary, we can assume that the a (cid:48) i and b (cid:48) j meet transversally in a single point. We thenobtain an admissible, multi-pointed Heegaard diagram H ρ = ( Σ , β ρ , α ρ , z , w ) by setting Σ = D ∪ − D (cid:48) , α i = a i ∪ a (cid:48) i and β i = b i ∪ b (cid:48) i .Note that in the definition of H ρ , we have interchanged the roles of the α and β -curves.Topologically, this has the effect of reversing the orientation of the ambient manifold — inthis case, the orientation of S . On the level of knot Floer homology groups, we have (cid:93) HFK ( H ρ ) ∼ = (cid:93) HFK ( − S , K ) ∼ = (cid:93) HFK ( S , m ( K )) .We call a homeomorphism ρ efficient if it minimizes intersections amongst the a (cid:48) i and b (cid:48) j within its mapping class. Such maps give rise to multi-pointed Heegaard diagrams whichwe also call efficient , and which are very close to being nice: they contain at most n − HKL ]. This trickconsists of two steps:( ) For each -sided bad region R in H ρ , stabilize H ρ as in Figure a by attaching a –handle to Σ with one foot in R and another in a region containing a z -basepoint.( ) Isotope the new β -curves as in Figure b by applying finger moves across the α -edges until reaching regions containing basepoints.The resulting diagram H nice after stabilizing and isotoping is nice [ LSV , Proposition . ]. . . The transverse invariant. Let L be a transverse link in ( S , ξ std ) which is braidedabout the standard (disk) open book decomposition for ( S , ξ std ) . To this braid, we associatea multi-pointed Heegaard diagram H ρ , as above. We call a diagram obtained in this way a braid diagram . P. LAMBERT-COLE AND D.S. VELA-VICK (cid:98) α i (cid:98) β i z i w i α i α i + (cid:98) t i ( a ) The multi-pointed Heegaard diagram af-ter stabilizing. Finger moves are applied asindicated by the (pink) dashed lines. (cid:98) α i (cid:98) β i z j w k β j β j + (cid:98) t i ( b ) The multi-pointed Heegaard diagram H nice , viewed from the “dual” perspectivewith β -curves appear visually straightened. F igure 2 . The Stabilization TrickIn the context of Heegaard Floer theory, braid diagrams first appeared in the work ofBaldwin, V´ertesi and the second author [ BVV ], and were used to establish an equivalenceof transverse invariants in knot Floer homology. Observe that the diagram H ρ supports adistinguished generator x [ H ρ ] = { t , . . . , t n − } , which is the union of the unique intersec-tions between the α and β -curves contained on the disk D ⊂ Σ . It was shown in [ BVV ]that the class [ x [ H ρ ]] ∈ (cid:91) HFK ( − S , L ) is an invariant of the transverse link L . This invariantis denoted t ( L ) and is commonly referred to as the BRAID invariant of transverse knots. TheMaslov and Alexander gradings of t ( L ) are given by M ( t ( L )) = sl ( L ) + A ( t ( L )) = sl ( L ) +
12 ,where sl ( L ) is the self-linking number of L . The self-linking number of the closure of an n -braid ρ is sl ( L ) = wr ( ρ ) − n where wr ( ρ ) is the writhe.In this paper, we work with the tilde version of knot Floer homology (cid:93) HFK ( − S , L ) instead of (cid:91) HFK ( − S , L ) . However, the generator x [ H ρ ] determines a class in (cid:93) HFK ( − S , L ) as well. Moreover, there exists a canonical projection map p : (cid:93) HFK ( − S , L ) → (cid:91) HFK ( − S , L ) with corresponding section s : (cid:91) HFK ( − S , L ) → (cid:93) HFK ( − S , L ) , such that s ( x [ H ρ ]) = x [ H ρ ] . RANSVERSE BRAIDS AND COMBINATORIAL KNOT FLOER HOMOLOGY . Identifying the transverse invariant Throughout this section, we assume that a transverse link L is given as the closure ofa braid ρ . By abuse of notation, we let ρ also denote an efficient homeomorphism in thecorresponding mapping class. . . The complexes C i , j . In this subsection, we inductively define a sequence of com-plexes C i , j and a sequence of chain-homotopy equivalences F i , j : (cid:103) CFK ( H ρ ) → C i , j . Thedifferentials on the complexes are defined analogously to those of (cid:103) CFK, except that thecounts associated to each Whitney disk are determined combinatorially. Similarly, the chainmaps are defined analogously to standard triangle maps, except with modified counts forWhitney triangles.Let H ρ be an efficient, multi-pointed Heegaard diagram for a braid closure. Via the“stabilization trick”, there is a sequence of multipointed Heegaard diagrams H ρ , H s , H , H , . . . , H N = H nice where( ) H s = ( Σ g , β ρ ∪ (cid:98) β , α ρ ∪ (cid:98) α , z , w ) is obtained from H ρ by g simultaneous stabilizationsin neighborhoods of the appropriate z –basepoints on the disk D ;( ) H = ( Σ g , β , α , z , w ) is obtained from H s by handlesliding across the (cid:98) β curves sothat each (cid:98) α k intersects the original -sphere along an arc from the region containing z k to the unique hexagon in annulus bounded by α k − and α k ; and( ) H i = ( Σ g , β i , α , z , w ) is obtained from H i − by an elementary isotopy of some (cid:98) β k across some α curve. In addition, by a small Hamiltonian isotopy we can assumethat each curve of β i intersects its corresponding curve in β , . . . , β i − transverselyin two points.Finally, let H (cid:48) = ( Σ g , β (cid:48) , α , z , w ) be a multi-pointed Heegaard diagram where thecurves of β (cid:48) are small Hamiltonian isotopes of the curves of β which each intersect theircounterpart in β , . . . , β N transversely in two points. qb y (cid:48) i y i α j (cid:98) β k F igure 3 . An elementary isotopy introduces two intersection points to α ∩ β .Each isotopy move from H to H nice introduces a pair of intersection points y i , y (cid:48) i to α ∩ β as in Figure . Let D i be the submodule spanned by generators with a vertex at y i or y (cid:48) i andno vertices at any y j or y (cid:48) j for j > i . Then, for each 0 ≤ i ≤ N , there is a decomposition ofmodules (cid:103) CFK ( H i ) = (cid:103) CFK ( H ) ⊕ D ⊕ · · · ⊕ D i . P. LAMBERT-COLE AND D.S. VELA-VICK
In particular, when i = N , we have (cid:103) CFK ( H N ) = (cid:103) CFK ( H nice ) . We additionally see that themodules D i have direct sum decompositions D i = D i ,1 ⊕ · · · ⊕ D i , n i where D i , j is spanned by two generators x i , j , x (cid:48) i , j satisfying x (cid:48) i , j = (cid:0) x i , j \ { y i } (cid:1) ∪ { y (cid:48) i } . Definean increasing sequence of submodules C i , j : = (cid:103) CFK ( H ) ⊕ i − (cid:77) k = D k ⊕ j (cid:77) m = D i , m for i =
1, . . . , N and j =
1, . . . , n i . Endow each intermediate module C i , j with the map d i , j ( x ) : = ∑ y ∑ φ ∈ π ( x , y ) µ ( φ )= n z ( φ )= n w ( φ )= N i , j ( φ ) · y where N ( φ ) is a modified count of representatives of the Whitney disk φ defined inductivelyas follows.First, we set d N , n N on C N , n N = (cid:103) CFK ( H nice ) to be exactly the differential on (cid:103) CFK ( H nice ) .Thus, define N N , n N ( φ ) : = (cid:99) M ( φ ) .where M ( φ ) denotes the moduli of pseudo-holomorphic representatives. For each pair ofintersection points y i , y (cid:48) i induced by the finger moves, there is a unique bigon –chain B i withcorners at y i and y (cid:48) i . This bigon determines a Whitney disk B i , j ∈ π ( x i , j , x (cid:48) i , j ) . Inductivelydefine the count N i , j by setting N i , j ( φ ) : = N i , j + ( φ ) + ∑ φ ∈ π ( x , y (cid:48) i , j ) φ ∈ π ( y i , j , x ) µ ( φ )= µ ( φ )= φ − B i , j + φ = φ n z ( φ )= n w ( φ )= n z ( φ )= n w ( φ )= N i , j + ( φ ) · N i , j + ( φ ) In defining the counts, we use the convention that ( i , n i ) ≡ ( i +
1, 0 ) .Next, we define chain-homotopy equivalences between (cid:103) CFK ( H ) and each C i , j by a mod-ified count of Whitney triangles. The three sets of curves β i , β , α , along with the basepoints,determine a Heegaard triple. There is a unique generator Θ i ∈ T β ∩ T β i of maximal Maslovgrading, which is closed. Define maps F i , j : (cid:103) CFK ( H ) → C i , j by the rule F i , j ( x ) : = ∑ y ∈ T α ∩ T β i ∑ ψ ∈ π ( Θ i , x , y ) µ ( ψ )= n z ( ψ )= n w ( ψ )= N i , j ( ψ ) · y The counts N i , j are again defined inductively. We set the map F N , n N to be the induced chainhomotopy equivalence from (cid:103) CFK ( H ) to (cid:103) CFK ( H nice ) . Thus, define N N , n N ( ψ ) : = (cid:99) M ( ψ ) RANSVERSE BRAIDS AND COMBINATORIAL KNOT FLOER HOMOLOGY We then define the triangle counts inductively using the formula N i , j ( ψ ) : = N i , j + ( ψ ) + ∑ ψ ∈ π ( Θ , x , x (cid:48) i , j ) φ ∈ π ( x i , j , y ) µ ( φ )= µ ( ψ )= ψ − B i , j + φ = ψ n z ( φ )= n w ( φ )= n z ( ψ )= n w ( ψ )= N i , j + ( φ ) · N i , j + ( ψ ) We now state the main result of this subsection.P roposition 3 . . For any i =
1, . . . ,
N and j =
1, . . . , n i , ( ) ( C i , j , d i , j ) is a chain complex, ( ) F i , j is a chain map, and ( ) F i , j : (cid:103) CFK ( H ) → C i , j is a chain-homotopy equivalence. This proposition follows easily from the following two lemmas. First, we use a standardfact in homological algebra to contract differentials and simplify the complex.L emma 3 . (Cancellation Lemma). Let ( C ⊕ D , d ) be a chain complex with differentiald = (cid:20) d CC d CD d DC d DD (cid:21) such that ( D , d DD ) is a contractible complex with null-homotopy H : D → D. Let ( C , d (cid:48) ) be thecomplex with twisted differential d (cid:48) = d CC + d CD ◦ H ◦ d DC . Then, the maps F : ( C , d (cid:48) ) → ( C ⊕ D , d ) and G : ( C ⊕ D , d ) → ( C , d (cid:48) ) defined byF : = Id C ⊕ H ◦ d DC G : = Id C + d CD ◦ Hare chain-homotopy equivalences.
Second, we establish the existence of a sequence of differentials to contract.L emma 3 . . For any i =
1, . . . ,
N and j =
1, . . . , n i , we have (cid:104) d i , j ( x i , j ) , x (cid:48) i , j (cid:105) = . We defer the proof of Lemma . until the end of this subsection. Using Lemmas . and . , we prove Proposition . .P roof of P roposition 3 . . All three statements are clearly true for i = N and j = n N since the differential and triangle map are defined by counting pseudo-holomorphic repre-sentatives.To prove the proposition, suppose by induction that all three statements are true for ( i , j + ) . Let d Di , j + denote the restriction of d i , j + to the submodule D i , j + . Then we havethat d Di , j + x i , j + = x (cid:48) i , j + by Lemma . . Consequently, ( D i , j + , d Di , j + ) is a contractible complexwith null-homotopy H i , j + defined by setting H ( x (cid:48) i , j + ) = x i , j + . Using the CancellationLemma (Lemma . ), we can contract D i , j + to obtain a twisted differential on C i , j and a P. LAMBERT-COLE AND D.S. VELA-VICK chain-homotopy equivalence G i , j : C i , j + → C i , j . This twisted differential is precisely d i , j and F i , j = G i , j ◦ F i , j + . All three statements are now clear for ( i , j ) . (cid:3) We now return to the proof of Lemma . . In preparation, we show that, like the countof pseudo-holomorphic representatives, if the count N i , j associated to a Whitney disk ortriangle is nonzero, then the corresponding domain in the Heegaard diagram is positive.L emma 3 . . Let φ denote a Whitney disk in the diagram H i , let ψ denote a Whitney triangle inthe multidiagram determined by β i , β , α , and let B i denote the bigon region in H i with corners at y i and y (cid:48) i . ( ) If the count N i , j ( φ ) is nonzero, then the -chain D ( φ ) + B i is positive. ( ) If the count N i , j ( ψ ) is nonzero, then the -chain D ( ψ ) + B i is positive. P roof . We begin by observing that if D ( φ ) + B i + is positive as a -chain in H i + , then D ( φ ) is positive as a -chain in H i .Also, both statements are clearly true for ( i , j ) = ( N , n N ) since both the differential andtriangle maps are determined by counting holomorphic representatives. If a holomorphicrepresentative exists, then by positivity of intersection, the -chain corresponding to theWhitney disk or triangle must be positive.Now, suppose by induction the statements are true for ( i , j + ) . If N i , j ( φ ) (cid:54) =
0, theneither N i , j + ( φ ) (cid:54) =
0, or there exist two domains φ , φ such that φ + φ − B i , j = φ and N i , j + ( φ ) and N i , j + ( φ ) are nonzero.In the first case, then clearly D ( φ ) + B i is positive by induction. In the second case,we know by induction that D ( φ ) + B i and D ( φ ) + B i are both positive. We will showthe stronger statements that D ( φ ) and D ( φ ) are positive. All α curves are adjacent to z -basepointed regions on both sides. Since n z ( φ ) =
0, this implies that the multiplicityof D ( φ ) can change by at most across any segment of α . In particular, pick two points b , q as in Figure . Then | n b ( φ ) − n q ( φ ) | ≤
1. Moreover, since φ has an outgoing cornerat y i , the multiplicities of φ must satisfy n b ( φ ) ≥ n q ( φ ) . However, since D ( φ ) + B i ispositive, this means that n q ( φ ) ≥
0. Thus n b ( φ ) ≥ n q ( φ ) ≥
0. Consequently, the domain D ( φ ) is positive. A corresponding argument shows that D ( φ ) is positive. As a result, D ( φ ) + B i = D ( φ ) + D ( φ ) is positive.A similar inductive argument proves the statement for triangle counts. (cid:3) We now finish the proof of Lemma . .P roof of L emma 3 . . First, the domain D ( B i , j ) = B i is clearly positive. If φ ∈ π ( x i , j , x (cid:48) i , j ) is any other Whitney disk, then D ( φ − B i , j ) is a periodic domain. Since the diagram H i is ad-missible, this periodic domain has both positive and negative multiplicities. Consequently, D ( φ ) is positive if and only if the negative component of D ( φ − B i , j ) is precisely B i . How-ever, this is not possible. If it was, then the boundary of the periodic domain must includethe α and β curves bounding B i . Yet, each β curve introduced by stabilization is not a linearcombination in H ( Σ k ) of the remaining α and β curves. Thus, it must show up with multi-plicity 0 in the boundary of any periodic domain. In turn, B i , j is the unique Whitney disk in RANSVERSE BRAIDS AND COMBINATORIAL KNOT FLOER HOMOLOGY π ( x i , j , x (cid:48) i , j ) with a positive domain and, by Lemma . , the only possible Whitney disk thatcould contribute to (cid:104) d i , j ( x i , j ) , x (cid:48) i , j (cid:105) . Therefore, it sufficies to check that N i , j ( B i , j ) = N N , n N ( B i , j ) = (cid:99) M ( B i , j ) = N i (cid:48) , j (cid:48) ( B i , j ) = N i (cid:48) , j (cid:48) + ( B i , j ) unless B i , j admits a decomposition B i , j = φ − B i (cid:48) , j (cid:48) + φ for some Whitneydisks with N i (cid:48) , j (cid:48) + ( φ ) = N i (cid:48) , j (cid:48) + ( φ ) =
1. If this occurs, then by Lemma . , the domains D ( φ ) , D ( φ ) are positive. However, since B i is an embedded bigon, this implies a fingermove introduces two new intersection points y i (cid:48) , y (cid:48) i (cid:48) on the α boundary arc of B i between y i and y (cid:48) i . But it is clear from Figure b that this never occurs in the isotopy. Consequently, thecount N i (cid:48) , j (cid:48) ( B i , j ) stays fixed at for all ( i , j ) ≤ ( i (cid:48) , j (cid:48) ) ≤ ( N , n N ) . In particular, N i , j ( B i , j ) = (cid:3) . . Transverse invariant. In this section, we identify the image of the transverse invari-ant t ( K ) under the chain homotopy equivalences defined in the previous section.Recall that x [ H ρ ] is the generator of (cid:103) CFK ( H ρ ) comprised of the distinguished intersec-tions t , . . . , t n − that lie on the portion of the original Heegaard surface coming from D .There are corresponding generators x [ H s ] in (cid:103) CFK ( H s ) and x [ H ] in (cid:103) CFK ( H ) specified bythe distinguished intersections along with the unique intersection points (cid:98) t k ∈ (cid:98) α k ∩ (cid:98) β k . Fi-nally, for i =
1, . . . , N , let T ( H i ) ⊂ (cid:103) CFK ( H i ) denote the subspace spanned by generatorswhose vertices along the curves α , . . . , α n − are precisely t , . . . , t n − . t i t (cid:48) i θ i α i β (cid:48) i β i z i − z i F igure 4 . All triangles with n z ( ψ ) = t i and θ i must have athird corner at t (cid:48) i .L emma 3 . . Let x [ H ρ ] , x [ H s ] , and x [ H ] be the distinguished generators in H ρ , H s , and H ,respectively, and let T ( H i ) be the subspace spanned by generators containing the distinguishedintersections { t i ∈ α i ∩ β i } . Then ( ) the chain homotopy equivalences f s : (cid:103) CFK ( H ρ ) → (cid:103) CFK ( H s ) and f h : (cid:103) CFK ( H s ) → (cid:103) CFK ( H ) satisfy f s ( x [ H ρ ]) = x [ H s ] f h ( x [ H s ]) = x [ H ] ;( ) for all i =
0, . . . ,
N, the subspace T ( H i ) is a subcomplex of (cid:103) CFK ( H i ) ; ( ) the chain homotopy equivalence f : (cid:103) CFK ( H ) → (cid:103) CFK ( H N ) satisfiesf ( x [ H ]) ⊂ T ( H N ) ; P. LAMBERT-COLE AND D.S. VELA-VICK ( ) for all i , j, the subspace T ( H i + ) ∩ C i , j is a subcomplex of C i , j ; and ( ) for all i , j, the chain homotopy equivalence G i , j : C i , j + → C i , j satisfiesG i , j ( T ( H i + ) ∩ C i , j + ) ⊂ T ( H i + ) ∩ C i , j P roof . Statements ( ) and ( ) follow from the same argument that x [ H ρ ] is closed. Anydomain with an outgoing corner at some vertex of x [ H ρ ] must cross one of the z basepointsand is therefore excluded from the differential. Statement ( ) now follows immediately fromStatement ( ) and the definition of the map G i , j in Lemma . .Next, there is an obvious identification f s of the generators of H ρ and H s and sincethe stabilizations are performed in neighborhoods of z –basepoints, the complexes (cid:103) CFK ( H ρ ) and (cid:103) CFK ( H s ) have identical differentials. Thus f s is a chain map that sends x [ H ρ ] to x [ H s ] .The chain homotopy equivalences f h and f are determined by counting triangles. However,any triangle with vertices at t i and θ i in Figure that misses the z basepoints must alsohave a third corner at t (cid:48) i . There is a unique such triangle, and it has a unique holomorphicrepresentative. This proves Statements ( ) and ( ). (cid:3) This suffices to show that F ( x [ H ]) is a scalar multiple of x [ H (cid:48) ] . Thus, the transverseinvariant is 0 if [ x [ H (cid:48) ]] =
0. To finish the proof of (main theorem), we need to prove theconverse as well.P roposition 3 . . The chain-homotopy equivalence F satisfiesF ( x [ H ]) = x [ H (cid:48) ] P roof . By Lemma . , we have that F ( x [ H ]) = ∑ ψ ∈ π ( Θ , x [ H ] , x [ H (cid:48) ]) µ ( ψ )= n z ( ψ )= n w ( ψ )= N ( ψ ) · x [ H (cid:48) ] Thus, we need to determine which domains in π ( Θ , x [ H ] , x [ H (cid:48) ]) contribute to the trian-gle map. The proof is similar to the proof of Lemma . . Specifically, there is a uniquepositive domain D ( ψ small ) , its contribution to F N , n N is exactly 1, and its contribution to eachsuccessive map F i , j can never deviate from 1.First, since β (cid:48) consists of small Hamiltonian isotopes of β , there is a obvious smalltriangle ψ small in π ( Θ , x [ H ] , x [ H (cid:48) ]) . It appear as an embedded domain for every triple β i , β , α for i = (cid:48) , 1, . . . , N . If ψ (cid:48) is any other triangle in π ( Θ , x [ H ] , x [ H (cid:48) ]) , then D ( ψ small − ψ (cid:48) ) is a multi-periodic domain. If the domain D ( ψ (cid:48) ) is positive, then its boundary mustinclude some curve (cid:98) α k with nonzero multiplicity. But each (cid:98) α k is linearly independent in H ( Σ g ) from the remaining α and β curves. This gives a contradiction so D ( ψ small ) is theunique positive domain.Secondly, in the triple β N , β , α , the domain appears and clearly has a unique holomor-phic representative. Thus N N , n N ( ψ small ) =
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