\mathbb{Z}_{2}-equivariant Heegaard Floer cohomology of knots in S^{3} as a strong Heegaard invariant

-equivariant Heegaard Floer cohomology of knots in as a strong Heegaard invariant

Sungkyung Kang Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK sungkyung.kang@maths.ox.ac.uk
Abstract.

The -equivariant Heegaard Floer cohomlogy of a knot in , constructed by Hendricks, Lipshitz, and Sarkar, is an isotopy invariant which is defined using bridge diagrams of drawn on a sphere. We prove that can be computed from knot Heegaard diagrams of and show that it is a strong Heegaard invariant. As a topolocial application, we construct a transverse knot invariant as an element of , which is a refinement of , and show that it is a refinement of both the LOSS invariant and the -equivariant contact class .

Key words and phrases:
Contact structure; Transverse knots; Heegaard Floer homology
2010 Mathematics Subject Classification:
57M27; 57R58
This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 674978).

1. Introduction

Suppose that a based knot in is given. Then we can represent as a bridge diagram on a sphere, and taking its branched double cover along the points where and the sphere intersect gives a Heegaard diagram of the branched double cover of along . This diagram admits a natural -action which fixes the basepoint and the ,-curves. From these data, Hendricks, Lipshitz, and Sarkar[eqv-Floer] gave a construction of the -equivariant Heegaard Floer cohomology , using their formulation of -equivariant Floer cohomology theory. They also proved that the isomorphism class of , which is a -module where is a formal variable, is an invariant of the isotopy class of the given knot . Also, the author proved in [Kang] that satisfies naturality and is functorial under based link cobordisms whose ends are knots.

Given these facts, it is natural to ask whether can be computed from bridge diagrams of drawn on closed surfaces of aribtrary genera, instead of spheres. In section 2, we will see that this is almost always possible, by proving the following theorem.

Theorem 1.1.

Let be a weakly admissible extended bridge diagrams representing a knot in , which has at least two A-arcs. Then .

Now, given such a fact, we can use it to compute from a weakly admissible knot Heegaard diagram of . To write it up clearly, choose a weakly admissible knot Heegaard diagram which represents a based knot . Add a pair of a small A-arc and a small B-arc connected to , whose interiors are disjoint; this gives a weakly admissible extended bridge diagram representing , which has at least two A-arcs. Then taking the branched double cover of the resulting extended bridge diagram and forgetting all basepoints except gives a Heegaard diagram of together with a -action. By Theorem 1.1, the -equivariant Heegaard Floer cohomology of this diagram is isomorphic to .

This construction implies that is a weak Heegaard invariant of , as defined in [Juhasz-naturality]. In section 3, we will see that the -equivariant Heegaard Floer cohomology, as a weak Heegaard invariant, satisfies certain commutativity axioms, thereby proving that it is actually a strong Heegaard invariant. Morover, we will also see that as a natural invariant calculated from bridge diagrams on a sphere is naturally isomorphic to as a strong Heegaard invariant; to be precise, we will prove the following theorem.

Theorem 1.2.

Consider the category  whose objects are based knots in and morphisms are self-diffeomorphisms of , and let

be the functor defined in Theorem 6.9 of [Kang]. Also, let

be the functor defined by considering as a strong Heegaard invariant. Then there exists an invertible natural transformation between and .

In section 4, we will see that any knot Heegaard diagrams representing a (based) knot in can be transformed, via isotopies and handleslides, to certain types of knot Heegaard diagrams, called very nice diagrams. Also, we will see that, from such a diagram, we can compute in a purely combinatorial way. As a result, we can remove a pair of an A-arc and a B-arc when computing from a knot Heegaard diagram, and thus extend Theorem 1.1 to full generality.

Theorem 1.3.

Let be a weakly admissible extended bridge diagrams representing a knot in . Then .

In section 5, we construct a new invariant associated to a knot in , whose isomorphism class is also an invariant of the isotopy class of , and prove that a version of localization isomorphism exists for . Finally, in section 6, we will construct an element associated to a transverse knot in the standard contact sphere , which depends only on the transverse isotopy class of , and see that it is a refinement of both the LOSS invariant defined in [LOSS-inv] and the -equivariant contact class defined by the author in [Kang].

Acknowledgement

The author would like to thank Andras Juhasz and Robert Lipshitz for helpful discussions and suggestions.

2. Equivariant Heegaard Floer cohomology and extended bridge diagrams

Definition 2.1.

Suppose that a Heegaard diagram is given. A based bridge diagram on is a 4-tuple , where is a finite subset of points in , are sets of simple arcs on , and , such that the following properties are satisfied.

  • For any , we have .

  • For any two distinct elements , we have , and the same statement holds for elements in .

  • For any and , the intersection is transverse.

  • For any , the two endpoints of are distinct and .

  • For any , there exists a unique element of which satisfies , and the same statement holds for .

Given a pointed bridge diagram on a Heegaard diagram , we call the elements of as A-arcs, the elements of as B-arcs, and as the basepoint. Note that is a pointed Heegaard diagram.

Suppose that a Heegaard diagram describes a 3-manifold . Then, given a based bridge diagram on , we can construct a based link lying inside as follows. Let be a Heegaard splitting of given by the Heegaard surface . Suppose that we call as the “outside” of and as the “inside” of . Then, we can isotope the A-arcs of slightly ourwards and the B-arcs of slightly inwards, while leaving the set fixed. Concatenating the isotoped arcs gives us a link , and the basepoint lies on , so that we get a based link in , which is uniquely determined up to (based) isotopy.

Definition 2.2.

Suppose that a Heegaard diagram describes a 3-manifold . We say that a based link in is represented by a based bridge diagram if the process described above gives a based link which is (based) isotopic to .

Proposition 2.3.

Suppose that a Heegaard diagram describes a 3-manifold . Then every based link in can be represented by a based bridge diagram on .

Proof.

Let be the Heegaard splitting of , induced by . Then for each , there exists a 1-subcomplex so that and . Since and are both 1-dimensional and is a 3-manifold, we can isotope so that , does not intersect and intersect transversely with . After further isotoping , we may assume that every component of admits a disk so that , and for any two distinct components of , we have . We can also assume that the disks does not intersect the family of compressing disks in , determined by the alpha- and beta-curves of , possibly after applying another isotopy to , while leaving the basepoint fixed. For each component , write where , i.e. is a simple arc on which is a projection of . By assumption, the arcs does not intersect the alpha- or beta-curves, and any two distinct arcs and do not intersect.

Now, after isotoping the arcs , we can assume that for any two curves and intersect transversely if , i.e. . Consider the following sets:

Then , and the based bridge diagram represents the given based link in . ∎

Definition 2.4.

An extended bridge diagram is a pair , where is a Heegaard diagram and is a based bridge diagram on . We write as and as . If and , the pointed Heegaard diagram will be denoted as . Also, the 3-manifold represented by is denoted as , and the based link in represented by is denoted as .

Given an extended bridge diagram , we have an associated 3-tuple , where is the 3-manifold represented by , and is a based link in , represented by . Obviously, given a 3-maniold together with a based link inside it, we have lots of extended bridge diagrams which represents it. In particular, we have a set of operations on extended bridge diagrams which leave the associated 3-manifold and based link fixed, which we will call as extended Heegaard moves. Also, we will call isotopies and handleslides involving ()-curves as ()-equivalences, and those involving A(B)-arcs and A(B)-equivalences. Finally, we will call ,-equivalences and A,B-equivalences as basic moves.

Isotopies

Given an extended bridge diagram, we can isotope its A-arcs, B-arcs, alpha-curves and beta-curves.

Handleslides of type I

Given an extended bridge diagram, we can replace an alpha(beta)-curve with another simple closed curve through an ordinary handleslide of knot Heegaard diagrams. Here, the handleslide region must not intersect any of the A/B-arcs.

Handleslides of type II

Given an extended bridge diagram, we can replace an alpha(beta)-curve with another simple closed curve if the following conditions are satisfied.

  • The curve does not intersect with any of the A(B)-arcs and the alpha(beta)-curves.

  • There exists an A-arc so that cobound a cylinder whose interior contains and does not intersect with any of the A-arcs, B-arcs, alpha-curves, and the beta-curves, except .

Handleslides of type III

Given an extended bridge diagram, we can replace an A(B)-arc with another simple arc if the following conditions are satisfied.

  • .

  • The interior of does not intersect with any of the A(B)-arcs and the alpha(beta)-curves.

  • There exists an alpha(beta)-curve so that bound a cylinder , whose interior does not intersect with any of the A-arcs, B-arcs, alpha-curves, and the beta-curves.

Handleslides of type IV

Given an extended bridge diagram, we can replace an A(B)-arc with another simple arc if the following conditions are satisfied.

  • The interior of does not intersect with any of the A(B)-arcs and the alpha(beta)-curves.

  • There exists an A(B)-arc such that bound a disk , whose interior contains and does not intersect with any of the A-arcs, B-arcs, alpha-curves, and the beta-curves, except for .

(De)stabilizations of type I

Given an extended bridge diagram , we can stabilize its Heegaard diagram at a point such that for any . The based bridge diagram remains the same.

(De)stabilizations of type II

Given an extended bridge diagram , where , choose an A-arc such that , and pick one of its endpoints, . Choose two distinct points lying in the interior of , so that the following conditions hold.

  • has three components , which are simple arcs on .

  • , , and .

  • and do not intersect with any of the B-arcs and beta-curves.

Then is again an extended bridge diagram.

Diffeomorphism

Given an extended bridge diagram and a diffeomorphism , we can apply on everything to get another extended bridge diagram.

Proposition 2.5.

Let be a Heegaard diagram which represent a 3-manifold . Any two based bridge diagrams on , which represent isotopic based links in , are related by isotopies, handleslides of type II, III, IV, and (de)stabilizations of type II.

Proof.

Choose a pair of a self-indexing Morse function and a Riemannian metric on , which induces the Heegaard diagram of . In the space of based link such that its basepoint lies on and it is transverse to at the basepoint, the subspace of based links which satisfy the conditions below is open and dense. Given any link in , its projection along the gradient flow of gives a based bridge diagram of on , up to stabilizations of type II.

  • The intersection is transverse.

  • The gradient vector field is nonvanishing on and transverse to .

  • For any flowline of whose endpoints lie on , the intersection is transverse.

  • For each bi-infinite flowline of , we have , and if the equality holds, we have .

  • The intersections of with the unstable manifolds of critical points of index and the stable manifolds of critical points of index are transverse.

We call the set as the set of points of codimension ; to prove the proposition, it suffices to classify the codimension singularities inside and show that they correspond to (compositions of) handleslides of type II, III, IV, and stabilizations of type II. It is easy to see that the codimension singularities in are given as follows.

  1. The link is tangent to at a point , such that and the order of tangency is .

  2. The link intersects transversely with either the stable manifold of a critical point of index or the unstable manifold of a critical point of index .

  3. There exists a flowline of which is tangent to at a point, such that the order of tangency is .

  4. There exists a bi-infinite flowline of such that and .

  5. The link is tangent to either the unstable manifold of a critical point of index or the stable manifold of a critical point of index , such that the order of tangency is .

  6. There exists three distinct points , different from the basepoint , and a bi-infinite flowline of such that .

The perturbations of the above singularities can be translated as the following compositions of extended Heegaard moves.

  1. A single stabilization of type II.

  2. A single handleslide of type III.

  3. The composition of two stabilization of type II and a handleslide of type II.

  4. The composition of a stabilization of type II and a handleslide of type IV.

  5. An isotopy.

  6. The composition of a stablilization of type II, a handleslide of type IV, and an isotopy.

Therefore we see that any two based bridge diagrams on which represent isotopic based links in are related by isotopies, handleslides of type II, III, IV, and (de)stabilizations of type II. ∎

Definition 2.6.

A based bridge diagram on a Heegaard diagram is simple if all A-arcs and B-arcs of lie in the same connected component of .

Theorem 2.7.

If two extended bridge diagrams which represent the same 3-manifold and isotopic based links, which are contained in a ball, they are related by extended Heegaard moves.

Proof.

Since the given link is assumed to be contained in a ball, for any Heegaard diagram , there exists a simple based bridge diagram which also represents the given link. Now, given any based bridge diagram on a Heegaard diagram , we know from Proposition 2.5 that we can apply isotopies, handleslides of type II, III, IV, and (de)stabilizations of type II to to reach . But then, we can regard the A-arcs and B-arcs together as a “big basepoint” and apply isotopies, handleslides, stabilizations and diffeomorphisms to the Heegaard diagram . The handleslides and stabilizations applied to corresponds to handleslides of type I and stabilizations of type I applied to the extended bridge diagram . Since any two Heegaard diagrams representing the same 3-manifold are related by isotopies, handleslides, stabilizations, and diffeomorphisms, the proof is complete. ∎

Remark.

By considering perturbations of Morse-Smale pairs on together with perturbations of the given link , and classifying all possible codimension singularities, we can remove the the assumption that our base link is contained in a ball, in Theorem 2.7. However, this observation is not necessary, as we will only consider knots and links in throughout this paper.

Now suppose that an extended bridge diagram , , is given, where the Heegaard diagram represents a 3-manifold and the based bridge diagram represents the isotopy class of a based link in . Then we construct a -tuple , which is defined as follows.

  • is the branched double cover of along .

  • , where is the branched covering map, and is defined similarly.

  • is the basepoint of the based link .

The -tuple , by construction, is a Heegaard diagram for the based 3-manifold , which is the branched double cover of the based 3-manifold along the link . The covering transformation of the branched cover induces an orientation-preserving -action on . We will say that is the branched double cover of along .

Proposition 2.8.

For any extended bridge diagram , the pointed Heegaard diagarm is weakly admissible if is weakly admissible.

Proof.

We will continue using the notations which we have used above. Consider the branched covering map . Then, for any connected component of , which does not contain the basepoint , we define its pullback as follows.

  • If is connected, it is a connected component of , so we define as .

  • If is disconnected, it consists of a -orbit of some connected component of , where acts as covering transformations. Denote that orbit as , where . Then we define as .

This definition can be extended linearly to give a group homomorphism

where for a point Heegaard diagram is defined to be the free abelian group of domains in which do not intersect the basepoint. The map clearly preserves periodicity.

Suppose that is weakly admissible and there exists a positive periodic domain . Then is also a positive periodic domain in . Using the proof of Lemma 4.2 in [Kang], we see that there exists a positive periodic domain such that . Since is assumed to be weakly admissible, we must have and thus . Since both and are positive, this implies , a contradiction. Therefore must be weakly admissible. ∎

We will now proceed to weak admissibilities of Heegaard triple diagrams and quadruple diagrams, which are perturbations of branched double covers of extended bridge diagrams. More precisely, the diagrams we will deal with are defined as follows.

Definition 2.9.

A 5-tuple is called an involutive Heegaard 5-tuple if the conditions below are satisfied.

  • is a branched double cover of a surface along a branching locus , such that .

  • The 4-tuples , , are pointed Heegaard diagrams.

  • There exist families of simple closed curves and families of simple arcs on such that is a Heegaard triple diagram, , , and are based bridge diagrams on the Heegaard diagrams , , and , respectively, and the branched double covers of along are , respectively.

A pointed Heegaard triple diagram is nearly involutive if it is given by a small perturbation of alpha-, beta-, and gamma-curves of some involutive Heegaard 5-tuple . We say that the pointed Heegaard triple diagram is the base of the nearly involutive triple diagram .

Proposition 2.10.

A nearly involutive Heegaard triple diagram is weakly admissible if its base is weakly admissible.

Proof.

The proof is the same as in Proposition 2.8, except that we are using triple diagrams instead of ordinary diagrams. Using the proof of Lemma 4.3 in [Kang], we see that the argument for ordinary diagrams can also be used for triple diagrams. ∎

Definition 2.11.

A pointed 6-tuple is an involutive Heegaard 6-tuple if any of the four 5-tuples given by excluding one out of four curve bases are involutive. A pointed Heegaard quadruple diagram is nearly admissible if it is given by a small perturbation of alpha-, beta-, gamma-, and delta-curves of some involutive Heegaard 6-tuple . If the bases of the triple diagrams given by exluding one out of four curve bases are given by a surface and curve bases on , then we say that the pointed Heegaard quadruple diagram is the base of the nearly involutive quadruple diagram .

Proposition 2.12.

A nearly involutive Heegaard quadruple diagram is weakly admissible if its base is weakly admissible.

Proof.

The proof is the same as in Proposition 2.10, except that we are now using the proof of Lemma 4.4, instead of the proof of Lemma 4.3, of [Kang]. ∎

The propositions 2.8, 2.10, and 2.12 tell us that, when we deal with nearly involutive diagrams, we do not have to care about their weak admissibility, as long as their base are weakly admissible. Hence, for the sake of simplicity, we will call a extended bridge diagram weakly admissible if the pointed Heegard diagram is weakly admissible. Note that this implies weak admissibility of .

Now, given an extended bridge diagram

such that is weakly admissible, we can apply the construction of [eqv-Floer] to the induced symplectic -action on the triple , where and is the genus of . What we get is the equivariant Floer cohomology

which is a -module in a natural way.

Lemma 2.13.

Given any two extended bridge diagram representing the same bridge link inside the same 3-manifold , such that and are weakly admissible and is contained in a ball, there exists a sequence of extended Heegaard moves which relates and , such that for every extended bridge diagram appearing in an intermediate step, the pointed Heegaard diagram is weakly admissible.

Proof.

From Proposition 2.8 and the proof of Theorem 2.7, we see that we do not have to consider based bridge diagrams of extended bridge diagrams. So we only have to care about extended Heegaard moves of pointed Heegaard diagrams. Since any two weakly admissible diagrams representing the same 3-manifold are related by isotopies, handleslides, stabilizations, and diffeomorphisms while preserving weak admissibility by Proposition 2.2 of [OSz-original], we are done. ∎

Given an extended Heegaard move of extended bridge diagrams whose source and target are both weakly admissible, we can associate it to a -module homomorphism between the corresponding -equivariant Floer cohomology, as defined in [eqv-Floer]. From Lemma 2.13, we know that any two weakly admissible extended bridge diagrams which represent the same based link inside the same 3-manifold, we have a map

which is defined as a composition of maps associated to extended Heegaard moves. The arguments used in the section 6 of [eqv-Floer] can be extended directly to show that the maps associated to extended Heegaard moves, except for stabilizations of type II, are isomorphisms.

Remark.

Here, we will assume that all stabilizations which we consider here occur near the basepoint, to ensure that they clearly induce isomorphisms of . In general, when we want to stabilize in a region which is not close to the basepoint, we can also associate it to an isomorphism by first performing it near the basepoint and then moving it via a sequence of handleslides. Of course, such an isomorphism is not unique; this problem will be resolved in the next section.

We now claim that, in some special cases, we can prove that stabilizations of type II induce isomorphisms. Note that, from now on, we will implicitly assume all extended bridge diagrams to be weakly admissible, by which we will mean that its base is weakly admissible; this is possible without loss of generality by Lemma 2.13.

Lemma 2.14.

Let be an extended bridge diagram, and let be the unique partition of such that every satisfies . Then applying a stabilization of type II to induces an isomorphism of equivariant Floer cohomology.

Proof.

Recall that, in the paper [eqv-Floer], the proof that stabilizations of bridge diagrams induce isomorphisms use equivariant transversality. That proof can be directly extended to our case, so if the -action on achieves equivariant transversality, then a stabilization of type II induces an isomorphisms.

For any domain of from a Floer generator to a generator , the Maslov index formula in the paper [cylindrical] reads:

where are point measures and is the Euler measure. Let be the branched double covering map with branching locus . Suppose that is -invariant. Then and are also -invariant, and thus we may assume without loss of generality that and , where are Floer generators in . By the assumption that every satsfies , the Maslov index of the domain , as defined in the proof of Proposition 2.8, is given as follows.

Therefore, the hypothesis (EH-2) in [eqv-Floer] is satisfied, and thus achieves equivariant transversality. ∎

Now we argue that, given any extended bridge diagram which represents a knot in , we can always adjust it to a position in which stabilizations of type II induce isomorphisms.

Definition 2.15.

An extended bridge diagram is said to be in a nice position if there exists an arc such that the following conditions hold.

  • .

  • The interior of does not intersect with any of the B-arcs and beta-curves.

Given an extended bridge diagram which is in a nice position, choose an arc as in Definition 2.15, and assume without loss of generality that is an A-arc. Let be an endpoint of such that . Choose two distinct interior points of such that the following condition is satisfied.

  • has three connected components , each of which is a simple arc, satisfying , , and .

Then, the pair is an extended bridge diagram, which represent the same based link in a same 3-manifold as .

Definition 2.16.

We define the above operation as special stabilization, i.e. applying a special stabilization to gives .

Lemma 2.17.

Let be a based knot in . Then any two extended bridge diagrams representing are related by isotopies, handleslides of type I, II, III, IV, stabilizations of type I near the basepoint, diffeomorphisms, and special stabilizations.

Proof.

We only have to prove that we can use special stabilizations instead of stabilizations of type II. Given any extended bridge diagram representing in , we can apply handleslide of type II, III, and IV to place it in a nice position. Since is a knot and thus has only one component, applying a stabilization of type II at any point has the same effect as applying a special stabilization and then moving the newly created pair of arcs to that point via isotopies and handleslide of type III. Therefore a stabilization of type II has the same effect as a composition of handleslides of type III, IV, and special stabilizations. ∎

Lemma 2.18.

Let be a based knot in . Suppose that an extended bridge diagram , which is in a nice position, represents in , and applying a special stabilization to gives another diagram . Then there exists an associated isomorphism:

Proof.

The extended bridge diagrams and , near the basepoint , are drawn in Figure 2.1. Since is the only B-arc/-curve which intersects the A-arc of , any Floer generator of can be written as

where is a uniquely determined Floer generator of .

Now assume that we are using almost complex structures of form , where is an almost complex structure on the Heegaard surface of (thus also of ) and is the genus of ; this is possible since such structures are enough to acheive transversality for all homotopy classes of Whitney disks. Since the regions of which contains on its boundary necessarily contains the basepoint , any holomorphic disks from to , where are Floer generators of , are actually holomorphic disks from to , as the point cancels out. Hence we have a homeomorphism

This implies that the map between the equivariant Floer complexes,

is an isomorphism. Therefore the induced map is also an isomorphism. ∎

Figure 2.1. Attaching a small A-arc and a B-arc at the point .

Now, given an extended bridge diagram which represents a knot in , we have a following commutative diagram of extended bridge diagrams, where the vertical arrows are stabilizations of type II and horizontal arroLews are either isotopies, handleslides of type I, II, III, IV, stabilizations of type I, or special stabilizations, and (and also ) are simple.

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