FLOER HOMOLOGY AND SINGULAR KNOTS

Transcription

1 FLOER HOMOLOGY AND SINGULAR KNOTS PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ arxiv: v3 [math.gt] 6 May 2009 Abstract. In this paper e define and investigate variants of the link Floer homology introduced by the first and third authors. More precisely, e define Floer homology theories for oriented, singular knots in S 3 and sho that one of these theories can be calculated combinatorially for planar singular knots. 1. Introduction Singular knots arise naturally in the skein theory of ordinary knots, see for example [4]. Moreover, they play a crucial role in the Khovanov-Roansky categorification of the HOMFLY-PT polynomial [3]. In this paper e define and investigate Floer homology theories for oriented, singular knots in S 3, hose Euler characteristic is the Alexander polynomial of the knot. The definitions of these theories are generaliations of knot and link Floer homology, [7], [11], [10]. Informally, one can think of a singular knot as an ordinary knot ith a finite set of double points. Hoever, for the purpose of skein theory, it is important to endo these objects ith some extra structure. To this end an abstract singular knot is a connected, trivalent graph ith a distinguished set of edges, called thick edges (and the remaining edges are called thin edges), hich satisfies an additional hypothesis: at each vertex, e require to of the edges to be thin, and the third to be thick. An oriented abstract singular knot is obtained by orienting all the edges in such a manner that at each vertex, both thin edges are oriented the same ay, hile the thick edge is oriented oppositely, see Figure 1(a). For an example of an oriented singular knot see Figure 1(b). Of course, not all abstract singular knots can be oriented in this ay; an example for a nonorientable abstract singular knot is shon in Figure 1(c). A singular knot, then, is a PL embedding of an oriented abstract singular knot in S 3. By contracting all the thick edges to points, e obtain an oriented knot ith a finite set of double points. As it is customary in knot theory, e ill mainly study singular knots through their generic projections. In fact, e consider projections hich have crossings only among thin edges. Singular knots arise naturally from projections of oriented knots as follos. Given a projection of an ordinary knot, e replace some of its crossings by inserting a PSO as supported by NSF grant number DMS AS as supported by OTKA T ZS as supported by NSF grant number DMS AS and ZS ere partially supported by the EU Marie Curie TOK program BudAlgGeo. 1

2 2 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ (a) (b) (c) Figure 1. Singular knots. (a) shos a thick edge, (b) illustrates the orientation convention, hile (c) is an example of a singular knot hich cannot be oriented. thick edge, in such a manner that the orientations of the thin edges are preserved. This is illustrated in Figure 2. Figure 2. Singulariing knot projections. The diagram shos ho to replace a positive/negative crossing by a thick edge in the projection.

3 FLOER HOMOLOGY AND SINGULAR KNOTS 3 Let K be an oriented, singular knot. The Alexander polynomial K (T) of K can be defined using the extension of Alexander polynomials of ordinary links via skein theory, as it is discussed in Section 3. In this paper e construct a bigraded homology theory associated to K hose graded Euler characteristic is determined by K (T). In fact, there are to variants of this homology theory, HFS(K) and HFS(K), and each splits as HFS(K) = s,d HFS d (K, s) and HFS(K) = s,d HFS d (K, s) here d (corresponding to the homological, or Maslov grading) is an integer, and s (the Alexander grading) is either an integer or a half-integer, depending on the number of thick edges of the singular knot. The group HFS(K) is a module over the polynomial ring F[U 1,...,U l ], here l + 1 denotes the number of thick edges of K (l 0) and F = Z/2Z. In fact, HFS(K) is gotten as the homology of a chain complex CFS(K) over this ring. The group HFS(K) is obtained by taking the homology of the complex CFS(K) gotten by specialiing all U i = 0 in CFS(K). Multiplication by U i drops Alexander grading by one, and it drops Maslov grading by to. Note that in the definition of HFS(K), the number of variables is one less than the number of thick edges; our construction treats one of these edges specially. Thus, these Floer homology groups depend on this choice of a distinguished thick edge on the singular knot; i.e. the resulting group is an invariant of the singular knot equipped ith a thick edge. The groups HFS(K) use all thick edges more symmetrically, and as such depends only on the singular link. See Theorem 2.4 for the invariance statement. The relationship beteen these invariants and the Alexander polynomial is spelled out in the folloing: Theorem 1.1. For the Floer homology theory HFS e have (1) χ( HFS (K, s)) T s = (1 T) l K (T), s here l + 1 denotes the number of thick edges in K. For HFS e have (2) χ(hfs (K, s)) T s = K (T). s A singular knot projection gives rise to a natural Heegaard diagram. Using such a diagram, and generaliing the notion of Kauffman states to singular knots (see Section 4 for details), e get a natural interpretation of a set of generators for the Floer homology groups in terms of these extended Kauffman states, similarly to [6, Theorem 1.2]. Theorem 1.2. Let K be a singular knot, and fix a decorated projection P of K. Consider the vector space C(K) over F generated by all generalied Kauffman states of P. There is a differential on C(K) hose homology calculates HFS(K).

4 4 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ A more precise statement is given in Section 4, here e also give an explicit formula for the bigrading associated to each generalied Kauffman state. We turn our attention to the Floer homologies of singular knots in the case here the singular knot K is planar, that is, K admits an injective projection to the plane. Theorem 1.3. Suppose that K is a planar singular knot. Then the group HFS(K) is determined by the Alexander polynomial K (T); indeed, the homology HFS d (K, s) is supported on the line 2s = d. It is more challenging to calculate HFS(K). In particular, e exhibit an example hich shos that HFS(K, s) can be non-trivial even hen its Euler characteristic vanishes. The present paper is organied as follos. In Section 2 e give the definition of HFS and HFS. In Section 3 e briefly discuss Alexander polynomials of singular knots. In Section 4 e give the state sum model for the Floer homology theory HFS of singular knots, leading us to the proofs of the theorems announced above. In Section 5 e sho an alternative Heegaard diagram for planar singular knots and do some calculations in some special cases. Most of the material described in this paper as discovered in 2004, hile the second author as visiting the Institute for Advanced Study. The invariants for singular knots described here (and some suitable modification of them) form the basis for a cube of resolutions description of knot Floer homology, described in [5], hich gives a construction of knot Floer homology quite similar in character to the Khovanov-Roansky categorification of the HOMFLY-PT polynomial [3]. A different construction of knot Floer homology for singular links has appeared very recently in [1], cf. also [13] Acknoledgements. The authors ish to thank the referee for a careful reading and extensive suggestions. 2. Definition of the Floer homology groups 2.1. Heegaard diagrams. Consider an oriented surface Σ of genus g, endoed ith to (g + l)-tuples of mutually pairise disjoint, embedded circles α = α 1,...,α g+l and β = β 1,..., β g+l (here l is a non-negative integer), chosen so that α 1,...,α g+l and β 1,...,β g+l determine handlebodies U α and U β ith U α = U β = Σ. If U α Σ U β = S 3, e call (Σ, α, β) a balanced Heegaard diagram for S 3. A self-indexing Morse function on S 3 ith l + 1 maxima and minima determines a balanced Heegaard diagram for S 3. A point p Σ α 1... α g+l β 1... β g+l determines a gradient flo hich connects some index ero and index three critical point. Indeed, given a collection of points = 1,..., a and = 1,..., b, each in Σ α 1... α g+l β 1... β g+l, e obtain an oriented graph Γ,, hich is oriented compatibly ith the flo along the edges corresponding to i, and oppositely for the edges corresponding to j. We suppose moreover that Γ, is a connected graph ith vertices of degree three. This

5 FLOER HOMOLOGY AND SINGULAR KNOTS 5 condition implies that each component {B i } l+1 i=1 of Σ β 1... β g+l has to points of type (denoted by i 1 and i 2), and one point of type (hich ill be denoted by i). A similar statement holds for the components {A j } l+1 j=1 of Σ α 1... α g+l. In this ay e obtain an oriented singular knot in S 3 by declaring the edges passing through 1,..., l+1 to be thick edges. Conversely, it is not hard to see that any singular knot K S 3 arises in this ay, that is, for any K there is a balanced Heegaard diagram compatible ith it. An example of such a Heegaard diagram ill be given in the next subsection Decorated projections and Heegaard diagrams. Fix a generic projection P of an oriented singular knot K. We also fix a generic initial point Q on the projection. We call this data a decorated knot projection for K. Given this data, e describe an associated Heegaard diagram, generaliing the one given in [6]. (Note that e have sitched here the roles of the α and β circles from the conventions from [6].) The handlebody U α is a regular neighborhood of the projection of K, and U β is its complement in S 3. Let X and Y be the regions in the projection hich contain the distinguished point Q. For each region in the complement of the projection other than X, e choose a corresponding β-circle given by the contour of the region. For each crossing in the projection e choose a circle α j supported in a neighborhood of the crossing as it is shon in Figure 3. For each thick edge, e choose a pair of α-circles α j and α j+1 hich are meridians for the to incoming arcs, and also an additional circle β i, hich meets only α j and α j+1 in to points apiece. We call β i an internal β-circle for the thick edge. This is illustrated in Figure 4. The diagram also shos ho to place the base points i and 1 i, 2 i near the thick edge. To complete the construction, finally e omit one of the internal β-circles. We ill take the convention that l+1 is the distinguished thick Figure 3. Heegaard diagram at a crossing. The position of the α circle (depending on the sign of the crossing) is shon.

6 6 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ edge. Moreover, in Section 4, e ill find it useful to choose a diagram here the initial point Q lies on a thin edge hich points into the distinguished thick edge, and that the internal β-circle for this thick edge is omitted. It is easy to see that the resulting balanced Heegaard diagram is compatible ith the given singular knot Definition of HFS(K) and HFS(K). Suppose no that K S 3 is a singular knot and (Σ, α, β,,) is a balanced Heegaard diagram compatible ith K. As usual (compare [8], [10]), e consider the (g + l)-fold symmetric poer Sym g+l (Σ) of the genus-g surface Σ, equipped ith the tori T α = α 1... α g+l and T β = β 1... β g+l in Sym g+l (Σ). Let π 2 (x,y) denote the set of homology classes of topological disks connecting the intersection points x,y T α T β. For p Σ α 1... α g+l β 1... β g+l and φ π 2 (x,y) e define n p (φ) as the intersection of φ ith V p = {p} Sym g+l 1 (Σ). We decompose the complement of the α i and β j into its components Σ α 1... α g+l β 1... β g+l = k C k. Given φ π 2 (x,y), the domain D(φ) associated to φ is the formal linear combination D(φ) = j n jc j, here here n j = n cj (φ) for any choice of c j C j. The homology class of φ is uniquely determined by its associated domain. We say that the domain D is nonnegative (riting D 0) if D = j n jc j ith n j 0 for all j. To functions and A : (T α T β ) (T α T β ) Z N : (T α T β ) (T α T β ) Z Figure 4. Heegaard diagram at a thick edge. The diagram describes the α-cirles and the internal β-circle corresponding to a thick edge, together ith the base points and to s.

7 FLOER HOMOLOGY AND SINGULAR KNOTS 7 can be defined by the formulas l+1 ( ) (3) A (x,y) = 2n i (φ) n 1 i (φ) n 2 i (φ) and i=1 l+1 N (x,y) = µ(φ) 2 n i (φ), here φ π 2 (x,y) is arbitrary, and µ(φ) is the Maslov index of φ π 2 (x,y). Lemma 2.1. The functions A and N are ell-defined, i.e. they are independent of the choice of φ π 2 (x,y). Moreover, given x,y, T α T β, e have that A (x,y) + A (y,) = A (x,) and N (x,y) + N (y,) = N (x,). Proof. Consider φ and φ π 2 (x,y). The difference D = D(φ) D(φ ) of their domains specifies a to-chain hose boundary is a sum of α- and β-circles. Since H 1 (Σ) is spanned by the images of the to g-dimensional subspaces spanned by the {[α i ]} g+l i=1 and {[β i]} g+l i=1, it follos that any such domain can be ritten as a sum of some D 1 and D 2, here i=1 l+1 l+1 D 1 = a i A i and D 2 = b i B i, i=1 here {A i } l+1 i=1 resp. {B i} l+1 i=1 are the components of Σ α 1... α g+l resp. Σ β 1... β g+l, and a i, b i Z. No, our combinatorial condition on the Heegaard diagram ensures that each component A i or B j contains to points of and one point of (corresponding to a thick edge). Since these latter points contribute to the sum in A ith multiplicity to, it follos readily that l+1 ( ) 2n i (φ) n 1 i (φ) n 2 i (φ) i=1 = l+1 i=1 i=1 ( ) 2n i (φ ) n 1 i (φ ) n 2 i (φ ) ; i.e. A is ell-defined. Well-definedness of N follos from standard properties of the Maslov index, see [10, Proposition 4.1]. Additivity of these quantities is straightforard. It follos from Lemma 2.1 that there are functions A: T α T β Z and N : T α T β Z, uniquely characteried up to an overall translation by the property that A(x) A(y) = A (x,y) and N(x) N(y) = N (x,y). It ill be useful to have the folloing

8 8 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ Lemma 2.2. Given x T α T β, there is a unique φ π 2 (x,x) ith n (φ) = n (φ) = 0. Proof. Notice first that the trivial class φ 0 π 2 (x,x) has the required property. Any other class in π 2 (x,x) differs from φ 0 by adding D 1 and D 2 as above. No the condition on n and n implies that for any a i in D 1 and b j in D 2 ith the property that A i B j contains some i or i e have that b j = a i. Using this repeatedly, together ith the fact that Γ, is connected, e see that D 1 + D 2 is ero, concluding the proof. We define the complex CFS(K) to be the chain complex freely generated over F[U 1,..., U l ] (ith F = Z/2Z) by the intersection points T α T β, endoed ith the differential (4) (x) = ( l ) # M(φ) U (n 1 (φ)+n 2(φ)) i i i y, y T α T β {φ π 2 (x,y) i=1 µ(φ)=1,n(φ)=0} here # M(φ) is the mod 2 count of holomorphic representatives of φ π 2 (x,y) up to reparametriation. The functions A and N defined above induce gradings on CFS(K), the Alexander and algebraic gradings, ith the convention that multiplication by U i drops A-grading by one, and preserves N. For the case of non-singular knots, knot Floer homology is bigraded, ith Alexander and Maslov degrees. The Maslov degree of [7] behaves differently from the grading N introduced here: multiplication by U i drops Maslov grading, hile it preserves N. In fact, e could define the folloing Maslov grading for the case of singular links by taking l+1 ( ) M (x,y) = µ(φ) + 2 n i (φ) n 1 i (φ) n 2 i (φ). i=1 This is analogous to the Maslov grading for knots, and in particular, ith respect to the induced grading on CFS(K), multiplication by any U i drops this grading by to. Note that e have the folloing easily checked relationship beteen the three gradings: N = M 2A. The complex CFS(K) is the chain complex e get by setting all U i = 0 (i = 1,..., l) in CFS(K). Equivalently, it is the free Abelian group generated over F by T α T β, endoed ith the differential (x) = y T α T β {φ π 2 (x,y) µ(φ)=1,n(φ)=0,n (φ)=0} # M(φ) y. Lemma 2.3. The maps and induce differentials on CFS and CFS hich drop the N-grading by one, and preserve the A-grading.

9 FLOER HOMOLOGY AND SINGULAR KNOTS 9 Proof. The sum defining is finite, according to Lemma 2.2. For, e argue that for fixed x and y, there are only finitely many φ π 2 (x,y) ith n (φ) = 0 and D(φ) 0. To see this, observe that the homology class of any φ ith n (φ) = 0 is uniquely determined by n (φ) (by Lemma 2.2), and moreover according to Lemma 2.1, the sum l+1 n 1 i (φ) + n 2 i (φ) i=1 is independent of the choice of φ (since it is A(y) A(x)). Since the coefficients in our theories are chosen in F = Z/2Z, the arguments of [10] apply. No the proof that 2 = 0 = 2 is an adaptation of [10, Lemma 4.3]. Note that, e have so far defined N and A only up to an overall additive constant. The indeterminacy in N can be easily removed ith the folloing observation. Suppose that e set all the U i = 1 in the definition of CFS (equivalently, consider the differential hich counts holomorphic disks ith n (φ) = 0, and ignore the reference points ). In this ay e obtain a chain complex hose homology is (up to an overall degree shift) isomorphic to H (T l ), see [10, Theorem 4.5]. Note that by Lemma 2.2, it follos that this diagram is admissible, so that the theorem applies. We choose N so that the homology group is isomorphic to H (T l ), ith a shift in the grading making the top-most homology supported in degree ero. The indeterminacy of A can be removed in an invariant manner using relative Spin c structures (compare [14]), though this is a slightly akard approach for the present applications. Rather, e ill leave the A-grading ell-defined only up to an overall shift, hich could be removed ith the help of the state sum formulae of Section 4. We no rite CFS(K) = d,s CFS d (K, s) and CFS(K) = d,s CFS d (K, s) here here CFS d (K, s) and CFS d (K, s) are generated by elements ith M = d Z and A = s. (Presently, the grading by s is ell-defined only up to an additive constant.) Theorem 2.4. Consider the homology groups HFS(K) = d,s HFS d (K, s) and HFS(K) = d,s HFS d (K, s) of the chain complexes ( CFS(K), ) and (CFS(K), ), here here d Z is an absolute integral grading and s (hich is in Z or Z+ 1, depending on the parity of the number of 2 thick edges) is a relative integral grading. Then, HFS(K) is an invariant of the singular knot K, together ith the distinguished thick edge; hile HFS(K) is an invariant of the singular knot.

10 10 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ Proof. By Morse theory it is knon that any to pointed Heegaard diagrams for a fixed singular knot can be connected by a sequence of (pointed) handleslides, isotopies, stabiliations and birth/death of index 0/1 (and 2/3) cancelling handles. Using the proof of [12, Lemma 1.1] e can avoid the appearance of cancelling 0/1 and 2/3 handles. For the remaining moves the proof of the invariance of the knot Floer homology groups follos the same lines as the proof of [10, Theorem 4.7] Other constructions. There are other variants of Floer homology for singular knots, hich generalie the constructions e describe here. For example, e could introduce a variable U l+1 for the final thick edge. The corresponding theory is the analogue for singular knots of HFK for knots (hile the construction HFS here plays the role of ĤFK). Indeed, this can be further generalied as follos: consider the chain complex C hich is freely generated by T α T β over the ring F[U 1,..., U m ], here m denotes the number of thin edges in our diagram, and endoed ith the differential: ) (x) = y T α T β {φ π 2 (x,y) µ(φ)=1,n(φ)=0} # M(φ) ( m i=1 U n i i (φ) The chain complex (CFS, ) described earlier can be thought of as the chain complex over the quotient ring here e set U e = U e, if e and e are to edges hich point into the same vertex; and also for some edge e 0, e set U e0 = 0. See [5] for related constructions. 3. Alexander polynomial of singular links State sum formulas for the Alexander polynomial of a smooth knot ere introduced in [2]. Our aim in this section is to recall the Alexander polynomial for singular links, and describe a state sum description for it, cf. also [4]. Proposition 3.1. There is a unique extension K (T) of the one-variable symmetried Alexander polynomial K (T) for non-singular, oriented links to singular, oriented links, hich is characteried by the skein relations K +(T) = K (T) + T 1 2 K o(t) K (T) = K (T) + T 1 2 K o(t), y. here here K o denotes the resolution of K at a singular point v, and K + and K denote the positive and negative resolutions at v (cf. Figure 5).

11 FLOER HOMOLOGY AND SINGULAR KNOTS 11 K + K K K o Figure 5. Resolutions at a thick edge. Proof. Uniqueness of such an extension follos from simple induction on the number of singular points, since K ± and K o have feer singular points than K. The fact that K (T) is ell-defined follos from a similar inductive argument together ith the fact that, because of the ordinary skein relation for the Alexander polynomial, the to equations above provide coherent results. From no on, the Alexander polynomial K (T) for a singular knot K ill be denoted simply by K (T). No e turn to the state sum description of K (T) for singular knots. To this end, first e recall the definition of Kauffman states for a singular link projection. Fix a decorated projection P for a singular link K ith initial point Q and contract the thick edges to points. Let Cr(P) and R(P) denote the set of crossings and the set of regions in the complement of the projection, resp. Let X and Y be the to regions hich contain the edge containing Q. A Kauffman state is a bijection x: Cr(P) R(P) X Y hich assigns to each crossing v Cr(P) one of the (up to) four regions hich meet at v. Let x be a Kauffman state and v a crossing in the projection. There is a local contribution Z v (x) defined according to hich quadrant x assigns to v, and the type of v (i.e., hether it is singular, positive, or negative). When v is non-singular, this local contribution takes values Z v (x) {1, ±T ±1 2 }, and hen v is singular, Z v (x) {0, 1, T 1 2 T 1 2 }. The precise rules are illustrated in Figure 6. This definition can be regarded as an appropriate extension of the local contributions for Kauffman states of ordinary links. In the case of ordinary links, the appropriate sum of the local contributions of Kauffman states provides the Alexander polynomial;

12 12 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ Proposition 3.2 generalies this fact to singular links. We note here that for our later purposes a refinement of Kauffman states (hich e ill call generalied Kauffman states) ill be introduced in Section 4. Proposition 3.2. Fix a decorated projection P of the oriented link K, let Cr(P) denote the set of crossings in the projection hile X(P) denotes its set of Kauffman states. Then the Alexander polynomial K (T) is calculated using the state sum formula (5) K (T) = x X(P) v Cr(P) Z v (x). Proof. The state sum formula for K o can be thought of as given by a state sum formula for states of K, here the local contribution at the resolved point contributes as in Figure 7. Since the state sum formula satisfies the skein relation of Proposition 3.1, and the identity of the proposition is knon to hold for a non-singular link, the formula of (5) for K (T) follos at once. T 1/2 0 T 1/ T 1/2 1/2 1/2 T T T 1/2 Figure 6. Local contributions. We illustrate here the local contribution function Z v (x) of a Kauffman state at a given crossing v. The left and right diagrams illustrate the case here the crossing is a non-singular point of K, hile the middle one illustrates the case here v is a singular point Figure 7. Local contributions at a resolved point.

13 FLOER HOMOLOGY AND SINGULAR KNOTS Kauffman states and the chain complex Our aim is to give a simple set of generators for a chain complex hose homology is isomorphic to HFS(K). These generators can be described concretely in terms of a suitable generaliation of Kauffman states, hich e describe presently. Fix a decorated projection P for a singular knot K ith initial point Q. We contract all the thick edges, so that they become crossings of the projection. Hoever, the four quadrants around a singular point v do not play equal roles. There are to quadrants, the side quadrants hich correspond to regions hich used to contain the thick edge in their boundary, hich e denote A v and C v. There are to remaing quadrants: the top quadrant B v, hich is pointed toards by the thick edge, and the bottom quadrant D v. We define four quadrants A v, B v, C v, and D v at the ordinary crossings analogously. We define a set, the set of Kauffman corners at v, for each crossing v of K, hich depends on hether v is an ordinary crossing or a contracted thick edge. If v is an ordinary crossing, the Kauffman corners are the four corners A, B, C, or D of the crossing v. If v is a singular crossing, the Kauffman corners correspond to A, C, D +, and D, here both D + and D belong to the bottom corner D. Let R(P) denote the set of regions in the complement of the projection. Let X and Y be the to regions hich contain the edge containing Q. Let Cr(P) denote the set of crossings of P (i.e. Cr(P) consists of the ordinary crossings and also the contracted thick edges). A generalied Kauffman state for a singular knot is a map x hich associates to each crossing v Cr(P) one of its four alloed Kauffman corners, ith the constraint that in each alloed region in R(P), there is a unique Kauffman corner in the image of x, compare [2]. (Note that this is a very mild generaliation of the notion from Section 3: there, e had four states corresponding to four quadrants, except one corner in the singular case contributed T 1 2 T 1 2; for our present purposes, it is convenient to think of both terms as corresponding to to different states D + and D.) Each Kauffman corner has a local Maslov grading M v, hich is ero on all Kauffman corners except for D (and D +, D at a singular point), here it is 1 according to the sign of the crossing. This choice of signs is illustrated in Figure 8. The Maslov grading of a generalied Kauffman state is the sum of local Maslov gradings over each crossing of K. (Notice that at the singular point no value at B is specified, since hen v is a singular point, B is not a Kauffman corner.) Similarly, each Kauffman corner has a local Alexander grading S v, defined as follos. If v is an ordinary positive crossing then S v (A) = S v (C) = 0, S v (B) = 1/2 and S v (D) = 1/2; hile if v is negative, then S v (A) = S v (C) = 0, S v (B) = 1/2 and S v (D) = 1/2; finally, if v is a singular crossing, then S v (A) = S v (C) = 0 and S v (D ) = 1/2, S v (D + ) = 1/2, see Figure 9. The Alexander grading S(x) of a Kauffman state x is given as a sum of local Alexander gradings. Fix a decorated projection for a singular knot K, ith the property that the distinguished edge Q is just underneath a singular crossing p. Consider next the associated

14 14 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ D D Figure 8. The local Maslov grading M v is illustrated at a crossing v. 1/ / /2 + D 1/2 D 1/2 01 1/2 Figure 9. crossing v. The local Alexander grading S v is illustrated at a Heegaard diagram for K described in Section 2.2. Leave out the internal β-circle β p associated to the singular crossing p hich lies above Q. Moreover, p corresponds to the special thick edge. It is easy to see that there is a Kauffman state associated to each x T α T β, determined as follos. Suppose that v is a singular crossing other than p, and let α j and α j+1 be the to circles supported in a neighborhood of the singular point v. Let β v be the corresponding internal β circle, and rite β A, β B, β C, and β D be the four β-circles corresponding to the four quadrants meeting at v. Our tuple x contains exactly one of the four points {a v, c v, d + v, d v } = (α j α j+1 ) (β A β C β D ), here a v β A, d ± v β D, and c v β C (note that the intersection cannot contain more points since β j = β v meets only α j and α j+1 ). We distinguish d + v and d v by the folloing convention: d + v lies on the same α-circle as c v. No, our Kauffman state is determined by x(v) = A v, D + v, D v, and C v, if x contains a v, d + v, d v, and c v respectively. The local picture at such a generic singular crossing is illustrated in Figure 10. At the singular point p, a similar but somehat simpler picture applies (the internal β-circle is missing, as is one of the β-arcs corresponding to a distinguished edge). The Kauffman state at an ordinary crossing is determined similarly, cf. [6].

15 FLOER HOMOLOGY AND SINGULAR KNOTS 15 B β B βa βc A C a v βv c v D d + v d v α α j j+1 β D Figure 10. The local picture of the Heegaard diagram near a singular point v. There are four unlabelled intersection points (the intersection points of β v ith α j and α j+1 ): to are hite and to are gray. The hite ones correspond to generators ith ǫ(v) = +1, the gray ones to those ith ǫ(v) = 1. Of course, each Kauffman state is associated to 2 l different intersection points of T α T β (here l + 1 denotes the number of singular points), distinguished by their coordinates on the internal β-circles. More specifically, the intersection points in T α T β are in one-to-one correspondence ith pairs x a generalied Kauffman state and ǫ: s(p) {±1}, here here s(p) denotes the set of singular points in the projection near hich e have an internal β-circle (i.e. all but the distinguished one, p). Suppose that (x, ǫ) and (x, ǫ ) represent to different intersection points x,x T α T β ith the same underlying Kauffman state, then e can find a homology class φ π 2 (x,x ) ith n (φ) = 0 and 2(n 1 v (φ) + n 2 v (φ)) = ǫ x (v) + ǫ x (v). In fact, this homology class is a union of disjoint bigons. In particular, each Kauffman state x is realied by a unique generator x T α T β hich has maximal Alexander grading among all generators realiing x, corresponding to the map ǫ(v) +1, for all v, cf. Figure 11. Our aim in this section is to establish the folloing precise version of Theorem 1.2: Theorem 4.1. Fix a decorated projection for a singular knot K ith the property that the distinguished edge Q is just belo a singular point. Let C d (K, s) be the free Abelian group generated by generalied Kauffman states ith Alexander grading s and Maslov grading d. There is a differential hich carries C d (K, s) to C d 1 (K, s), ith : C(K, s) C(K, s) H d (C (K, s), ) = HFS d (K, s).

16 16 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ + D D Figure 11. Alexander grading maximiers. For each Kauffman corner illustrated in the top ro, e have dran its corresponding Alexander grading maximiing intersection point belo it. Before giving the proof, e establish some lemmas. Indeed, e find it convenient to focus first on the case here K is a planar singular link (hich is in fact the case here Theorem 4.1 is the most valuable) Planar singular links. We analye the Heegaard diagram in the case here K is a planar singular link (that is, admits an injective projection to the plane), establishing a special case of Theorem 4.1 in this case, from hich Theorem 1.3 follos. First, e establish several lemmas. Lemma 4.2. Let K be a planar singular link. Let x and y be to generalied Kauffman states, and let x,y T α T β unique Alexander grading maximiing intersection points representing them. Then, the difference beteen the Alexander gradings of x and y (in the sense of Equation (3)) coincides ith the difference beteen the Alexander gradings S(x) and S(y) of the Kauffman states x and y. Proof. Given x and y, it suffices to construct a particular homology class φ π 2 (x,y) hich satisfies l+1 (6) S(x) S(y) = (2n i (φ) n 1 i (φ) n 2 i (φ)). i=1 Indeed, it suffices to consider the case here there are no D corners in x or y, in vie of the folloing observation. Consider first the special case here x and y agree at all but n corners, here x is assigned D + and y is assigned D. In this case, it is easy to find

17 FLOER HOMOLOGY AND SINGULAR KNOTS 17 a locally supported homology class φ π 2 (x,y) ith n i (φ) = n, n i (φ) 0, as illustrated in Figure 12. Suppose then that x and y contain no D corners. We construct a curve γ hich connects x to y, as in Figure 13. This curve γ is a (possibly disconnected) closed path constructed from arcs ithin the α and β-circles. In fact, if x has components x i and y has components y i, then γ is constructed from arcs in α i from x i to y i and arcs in β j from y j to x k. No, order the edges {e i } in the order they are encountered in the singular knot, starting from Q. Let v be some crossing here e i and e i+1 meet. Fix reference points T i on the Heegaard diagram, placed on top of the i th edge, see the loer left picture in Figure 13. We can dra an arc ǫ i from T i to T i+1, hich crosses only one of the α-circle hich is the meridian for the i th edge, and none of the other circles. Similarly, let η i be the short arcs going from v to 1 v and 2 v (inside the disk bounded by β v). Then, A(x) A(y) = #(η i φ) since i ǫ i is a closed curve. We claim that = #((ǫ i + η i ) φ), (7) #(ǫ i + η i ) φ = S v (x) S v (y) 1 2 ( Wx(v) (K) W y(v) (K) ), here here W r (K) denotes the inding number of K around a point in the region r. We see this as follos. The only part of φ hich intersects ǫ i + η i lies on the internal β-circle. Since the intersection number of ǫ i + η i ith the internal β-circle is ero, it suffices to verify Equation (7) replacing φ ith any arc connecting x to y on the internal β-circle. Thus, e can verify Equation (7) by considering the various cases of x and y (locally, about each singular point), as illustrated in Figure 13. D D 01 Figure 12. Domains for comparing D + and D. We have illustrated on the right a domain φ hich connects the Alexander grading maximiing generator of D + ith that of D. All multiplicities are +1, 1, or 0; regions ith local multiplicity +1 are indicated by hatch marks from upper right to loer left, hile regions ith multiplicity 1 are indicated by the other hatching.

18 18 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ No, summing Equation (7), e see that A(x) A(y) = S(x) 1 2 W x(v) (K) S(y) 1 2 But observe that v Cr(P) v Cr(P) W x(v) (K) = r R(P) X Y W r (K) is independent of the Kauffman state x. This finishes the proof. v Cr(P) W y(v) (K). Definition 4.3. Let K be a planar singular knot projection. To generalied Kauffman states x and y are said to be equivalent if x(v) {A v, Dv } if and only if y(v) {A v, Dv }. Let x be a generalied Kauffman state. This determines an associated subgraph of the knot projection as follos. At each crossing v, if x(v) {A v, Dv }, then e remove the loer left edge from the projection; if x(v) {C v, D v + }, e remove the loer right edge from the crossing. We call the associated subgraph Γ x of the projection the pruning associated to the Kauffman state. Thus, to Kauffman states are equivalent if and only if they induce the same pruning. Note that the pruning is a graph, ith exactly one in-coming edge at each vertex, and at most to out-going ones. D + + D B j+1 Tj+1 Ti+1 Bi+1 Bi Ti B j Tj Figure 13. Verifying Equation (7). We verify that equation, using arcs connecting the black dot (representing x) to the hite dot (representing y) supported on the internal β-circle.

19 FLOER HOMOLOGY AND SINGULAR KNOTS 19 Lemma 4.4. The pruning associated to a Kauffman state for a planar singular knot is a connected graph. Proof. Suppose that Γ x is not connected. We argue that Γ x must contain some cycle. This is seen by taking some edge e hich is in a different path component from the initial point, and continuing backards through e. Since each vertex in Γ x has a unique incoming edge, this process can be continued; so the component through e must contain a cycle. No, e consider some closed circuit C in Γ x in a path component disjoint from the distinguished edge Q. The circuit C can be thought of as enclosing a region of the projection hich does not contain the distinguished edge. We restrict our Kauffman state to. The restriction of our projection to gives the region, hich is topologically a disk, the structure of a CW complex, here the vertices correspond to the V crossings, and edges corresponding to the E arcs in, and hich divide into F faces. There are four kinds of vertices in the one-complex of : bivalent ones (ith one in-coming and one out-going edge) hose number is D, vertices ith to in-coming and one out-going edge hose number is T 1, vertices ith one in-coming and to out-going edges hose number is T 2, and four-valent ones (to in-coming, and to out-going edges) hose number is W, so that V = D + T 1 + T 2 + W. Note that all vertices counted in D, T 1, and T 2 occur on the boundary of. By counting edges, e can verify that T 1 = T 2. In fact, the total number of edges is given by E = D + 3T 2 + 2W. But since is a disk, its Euler characteristic is one, so e get the relation that F T 2 W = 1. Since each face in is occupied by a Kauffman corner (e are using here the fact that Q is not contained in ), the restriction of our Kauffman state x to demonstrates that F T 1 + W (at each vertex v counted in D or T 2, x(v) is not a face of, hereas at each vertex v in W, x(v) is a face of, hile at each vertex v in T 1, x(v) might or might not be a face of ), contradicting T 1 = T 2 and F T 2 W = 1. This has the folloing easy consequence: Lemma 4.5. If x and y are generalied Kauffman states for a planar singular knot K hich are equivalent, then x and y coincide. Proof. We construct the folloing subset P of the complement of Γ x. Suppose that x and y differ at a crossing v. Then, of course there is a different crossing ith the property that y() and x(v) are assigned to the same region. We then connect y() and x(v) along some arc in the complement of the knot projection. Next, e connect x(v) to y(v) by an arc hich crosses one of the four edges of our knot projection; but that is precisely the edge removed to obtain Γ x. We continue this procedure. In this manner, e construct a collection of closed curves P. If P is non-empty, let R be any connected component of P. It is easy to see that R divides the plane into

20 20 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ to regions, both of hich contain points in Γ x. But this contradicts the fact that (by Lemma 4.4) Γ x is connected. We no have the ingredients required to establish Theorem 4.1, at least in the case hen K is a planar singular link. Proposition 4.6. Theorem 4.1 holds for planar singular links; i.e. if K is a planar singular link, let C d (K, s) be the free Abelian group generated by generalied Kauffman states ith Alexander grading s and Maslov grading d. There is a differential hich carries C d (K, s) to C d 1 (K, s), ith : C(K, s) C(K, s) H d (C (K, s), ) = HFS d (K, s). Proof. If ǫ and ǫ differ at a single singular point v, then the homology class φ clearly admits a single holomorphic representative up to reparametriation. In fact, by placing basepoints at all the other regions of our Heegaard diagram, e obtain a filtration of the chain complex CFS hose E 0 term counts only these short differentials; i.e. its differentials are given by tensor product of the space of Kauffman states ith l chain compexes of the form U F[U 1,...,U l ] i F[U1,...,U l ]. It is easy to see that the E 1 term, no, is simply the free F-module generated by the Kauffman states. Indeed, this homology is carried by the pairs (x, ǫ ), here ǫ (v) = 1 for each v. It remains to identify the Alexander and Maslov gradings of the intersection points (x, ǫ ) ith the corresponding gradings for their underlying Kauffman states. This statement for the Alexander grading is an immediate consequence of Lemma 4.2. For the statement about the Maslov gradings, it suffices to sho that the algebraic grading N of any of the Alexander grading maximiing generators vanishes identically. To this end, let x be a generator and x its underlying Kauffman state. We claim that there is a ne Heegaard diagram, obtained by moving the central β-circles, each across exactly one of the to j i for fixed j, so as to cancel to of the intersection points of this β ith one of the to α-circle meridians, depending on the value of x(v), as illustrated in Figure 14. Let γ v denote the isotoped image of β v. Note that the γ v are chosen so that the intersection point x persists into the ne diagram. Indeed, the generators for the ne diagram no correspond to pairs (y, ǫ), here here y is some Kauffman state equivalent to x, and ǫ: s(p) {±1}. According to Lemma 4.5 it follos at once that the generators for the ne diagram correspond simply to maps ǫ: s(p) {±1}, since they all have the same underlying Kauffman state. We claim that this is an admissible Heegaard diagram for S 3 ith l basepoints (supplied by ). To see admissibility, e proceed as follos. Let Π be a periodic domain ith multiplicity m near some singular

21 FLOER HOMOLOGY AND SINGULAR KNOTS 21 + D D Figure 14. Isotopies of the standard diagram. Given the generator x, e consider the ne diagram obtained by isotoping the central β-circle as indicated by the light dotted line. Note that the ne diagram for A and D coincide, hile the diagram for C and D + coincide. point v. Let u be the preceding vertex to v in the pruning Γ x. If Π has only nonnegative multiplicities, then its local multiplicity near u must be greater than or equal to m. Iterating this, e can go back to the root of the pruning, in vie of Lemma 4.4, here the local multiplicity in turn must equal ero. This shos that our periodic domain must vanish identically. m c 0 m+c 0 Figure 15. Verifying admissibility. We have illustrated here the Heegaard diagram near to vertices, along ith part of a periodic domain. (We have suppressed the basepoints, as they no longer play a role.) If the periodic domain has local multiplicity m near the vertex v, and local multiplicity c in the small region, then its local multiplicity near the preceding vertex u in the pruning must be m + c.

22 22 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ Next, note that the isotopy, of course, changes the Floer homology; but it leaves invariant the theory here all U i = 1 for i > 1 (inducing an isomorphism beteen N-graded theories). Indeed, recall [9] that the isomorphism induced on homology by isotopies can be thought of as induced by a chain map hich counts holomorphic triangles. Moreover, the count of holomorphic triangles clearly takes the generator corresponding to (x, ǫ), to the generator corresponding to the same ǫ (respecting N-gradings). The result follos. This immediately gives a proof of Theorem 1.3: Proof. [of Theorem 1.3.] Applying Proposition 4.6, e see that the differential actually must vanish identically, since the state sum ensures that for fixed Alexander grading, the generators of the chain complex have fixed Maslov grading The general case of Theorem 4.1. The discussion above can be generalied to the case of (non-planar) singular knots, as ell. For example, e have the folloing generaliation of Lemma 4.2: Lemma 4.7. Let K be a singular link. Let x and y be to generalied Kauffman states, and let x,y T α T β unique Alexander grading maximiing intersection points representing them. Then, the difference beteen the Alexander gradings of x and y (in the sense of Equation (3)) coincides ith the difference beteen the Alexander gradings S(x) and S(y) of Kauffman states x and y. Proof. We argue as in the proof of Lemma 4.2, establishing Equation (7) in the presence of non-singular intersection points. This time, it is the central α-circle hich meets the curve ǫ i. Its total intersection number ith this curve is ero, so it sufficies to verify Equation (7) by replacing φ ith arcs in the central α-circle hich connect the to generators x and y, compare Figure 16. This is straightforard to verify, and the previous proof goes through. Proof. [of Theorem 4.1] Proceed as in the proof of Proposition 4.6 to see that there is a chain complex generated by Kauffman states. Lemma 4.7 verifies the identification beteen the Alexander gradings coming from the Heegaard diagram ith the state sum formula. It remains to verify that the Maslov grading for each Alexander grading maximiing generator x corresponding to a given Kauffman state x is given by the state sum formula; indeed, since e have already verified the corresponding statement for the Alexander gradings, it suffices to verify the corresponding statement for the N-grading (hich has the advantage that it is independent of the placement of the i basepoints). Proceeding as before (as indicated in Figure 14), e isotope β v across one of the basepoints i j at

23 FLOER HOMOLOGY AND SINGULAR KNOTS 23 B i+1 B j+1 T j+1 T i+1 T i T j B j Bi Figure 16. Verifying Equation (7) in the presence of nonsingular crossings. The black dot represents x, the hite represents y. For negative crossings, rotate the diagram. each singular crossing v to get a ne circle. In fact, e can equivalently vie this as an isotopy of one of the α-circles intersecting β v, replacing it ith a ne circle γ v disjoint from β v. Consider next a non-singular crossing v. If x(v) is of type A or C, e replace the α-circle by a ne circle γ v hich is a meridian for the corresponding in-coming edge. If x(v) is of type D, e replace its corresponding β-circle by the meridian γ v for either of the to in-coming edges. Finally if x(v) is of type B, e replace the α-circle by a ne circle γ v supported locally near the crossing pictured, as shon by Figure 17. In this manner, e obtain a ne diagram (Σ, γ, β,,), equipped ith a set S = T γ T β, from hich e have a map to the Kauffman states for K. Indeed, elements of S map to the Kauffman states for the projection of another knot K 0 hich is obtained by resolving all the crossings here x(v) = B. Let x 0 denote the induced Kauffman state on K 0. It is easy to see that any other intersection point y of T γ T β induces a Kauffman state y hich associates B or D to each vertex v here x(v) = B, and hence it can be restricted to K 0. In fact, y 0 is equivalent to x 0, and hence by Lemma 4.5 x 0 = y 0. From this, it follos easily that x = y. Indeed, Lemma 4.4 shos that the γ circles divide Σ into l+1 components. Since the span of the γ i is contained in the span of the α i, it follos that these to spans coincide, and that (Σ, γ, β,) is a Heegaard diagram for S 3. No, given generator x T α T β (maximiing Alexander grading among all intersection points corresponding to the Kauffman state x), e have exhibited a ne

24 24 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ γ γ α α Figure 17. A generator Θ. The black dots represent the generator Θ T γ T α. At each vertex, the γ- and α-circles meet in to points, one of hich e colored black, the other hite. For a positive crossing, the black has N-grading one greater than the hite generator; this can be verified using the obvious bigon (minus disks) going from black to hite. Similarly, at a negative crossing, the N-grading of the hite generator is one greater than the N-grading of the black one. Heegaard diagram, equipped ith a corresponding Alexander grading maximiing generator x T α T γ, hich clearly has N-grading equal to ero. In fact, T α T γ also has a minimal number of intersection points. Let Θ denote its N-maximiing generator. If n denotes the number of negative crossings v here x(v) = B (and x is the Kauffman state corresponding to x), and p denotes the number of such positive crossings, then e claim that there is an obvious ψ π 2 (Θ,x,x ) here gr(θ ) = gr(θ) n, and µ(ψ) = p, cf. Figure 18. The map induced by counting holomorphic triangles preserves gradings, in the sense that N(Θ ) + N(x) µ(ψ) = N(x ), from hich it follos no that N(x) = n p. Comparing ith Figures 8 and 9, e have verified that N(x) = M(x) 2S(x). The proof of Theorem 1.1 is an immediate corollary of Theorem 4.1: Proof. [of Theorem 1.1.] Equation (2) is an immediate consequence of Theorem 4.1 and Proposition 3.2. Equation (1) follos by a simple comparison of the chain complexes: each generator x for CFS corresponds infinitely many generators for U n U n l l indexed by l-tuples

25 FLOER HOMOLOGY AND SINGULAR KNOTS 25 Figure 18. Triangles. We denote the generator Θ by a grey dot; x T α T γ and x T α T β. The triangle on the left (corresponding to the positive crossing) has local coefficent 1 (and indeed Maslov index equal to 1) hile the one on the right has positive local multiplicity (and Maslov index equal to ero). of non-negative integers (n 1,..., n l ). Each such generator occupies Alexander grading A(x) l k=1 n k and N-grading equal to the N-grading of x. Equation (1) follos. According to Theorem 1.3, calculating HFS(K) for a planar knot is equivalent to computing its Alexander polynomial. This can be efficiently done either using the state sum formula K (T) = Z v (x) x K(P) v Cr(P)) e have discussed in Section 3, or the skein relations K +(T) = K (T) + T 1 2 K o(t) K (T) = K (T) + T 1 2 K o(t), e have used in the definition of the Alexander polynomial for singular links. 5. Some calculations 5.1. Planar singular links. Recall that a singular link K is called planar if it admits an injective projection to the plane. For such links a simple Heegaard diagram of genus

26 26 PETER OZSVÁTH, ANDRÁS STIPSICZ, AND ZOLTÁN SZABÓ ero can be given in the folloing ay. Fix a planar singular link K, consider an injective projection, contract all its thick edges to singular points and take the α- and β-curves at every crossing as it is instructed by Figure 19. Note that an α-curve corresponds to each thick edge and the to outgoing thin edges, hile a β-curve corresponds to a thick edge and to incoming thin edges. By taking these curves for all the crossings and then deleting an (arbitrary) α- and β-curve from the collection, e get a Heegaard diagram of a singular link. Proposition 5.1. The resulting Heegaard diagram is compatible ith the given planar singular link K. We can distinguish coordinates of an intersection point x = (x 1,...,x l ) T α T β according to hether x i corresponds to a thin or thick edge in the diagram it is reflected by the fact hether x i is near a base point of type or of type. Since near a fixed or the α and β curves intersect each other in to points, e can group our intersection points into groups of cardinality 2 l An example. We ill illustrate the above principle by an example. This example also shos that the Floer homology theory HFS is not determined naively by the Alexander polynomial of a planar singular knot. By taking the (3,3) torus link and singulariing its natural projection e get the singular knot K depicted by Figure 20. The planar Heegaard diagram corresponding to this projection is shon by Figure 21. The diagram also indicates (ith dashed lines) the α- and β-curves hich e delete according to the algorithm for constructing the diagram from the projection. The Alexander polynomial of the singular knot K given above can easily be computed from the state sum formula, giving K (T) = T T T 1 + T 2. α β Figure 19. Heegaard diagram of a planar singular link. The crossing on the left (obtained by contracting a thick edge) is replaced by the piece of Heegaard diagram on the right.