Formality of Floer complex of the ideal boundary of hyperbolic knot complement
This is a sequel to the authors’ article [BKO]. We consider a hyperbolic knot in a closed 3-manifold and the cotangent bundle of its complement . We equip a hyperbolic metric with and the induced kinetic energy Hamiltonian and Sasakian almost complex structure with the cotangent bundle . We consider the conormal of a horo-torus , i.e., the cusp cross-section given by a level set of the Busemann function in the cusp end and maps converging to a non-constant Hamiltonian chord of at each puncture of , a boundary-punctured open Riemann surface of genus zero with boundary. We prove that all non-constant Hamiltonian chords are transversal and of Morse index 0 relative to the horo-torus . As a consequence, we prove that unless and an -algebra associated to is reduced to a noncommutative algebra concentrated to degree 0. We then prove that the wrapped Floer cohomology with respect to is well-defined and isomorphic to the Knot Floer cohomology that was introduced in [BKO] for arbitrary knot . We also define a reduced cohomology, denoted by , by modding out constant chords and prove that if for some , then cannot be hyperbolic.
Key words and phrases:Hyperbolic knots, Knot Floer algebra, horo-torus, formality, totally geodesic triangle
- 1 Introduction
- 2 Definition of Knot Floer algebra in [BKO]
- 3 Preliminary on hyperbolic 3-manifold of finite volume
- 4 Cotangent bundle of hyperbolic knot complement
- 5 The kinetic energy Hamiltonian and hyperbolic geodesics
- 6 Second variation and Morse index calculation
- 7 Perturbed Cauchy-Riemann equation and structure
- 8 The maximum principle and -estimates
- 9 Formality of algebra associated to hyperbolic knot
- 10 Comparison with Knot Floer cohomology
- 11 Reduction of the classification problem to 2 dimension
- A Computation of : Proof of Proposition 11.4
- B Proof of Lemma 9.9
The symplectic idea of constructing knot invariants using the conormal lift of a knot (or link) in as a Legendrian submanifold in the unit cotangent bundle has been explored in symplectic-contact geometry, especially exploited by Ekholm-Etnyre-Ng-Sullivan [EENS] in their construction of knot contact homology who proved that this analytic invariant recovers Ng’s combinatorial invariants of the knot [Ng]. It has been also observed (see [ENS] for example) that the data of the knot contact homology can be obtained from a version of wrapped Fukaya category on the ambient space, the symplectization of the unit cotangent bundle or of an open subset thereof.
In [BKO], the authors considered the knot complement of arbitrary orientable closed 3-manifold directly, and constructed its associated Fukaya category on it. We emphasize that the base space is non-compact. We take a tubular neighborhood of and consider its boundary . We define a cylindrical adjustment of the induced metric of a smooth metric of . (See Section 10 for the precise definition thereof.) Then the construction in [BKO] associates an algebra
We denote the associated cohomology by
It was shown in [BKO] that this cohomology does not depend on the choices of smooth metric on , of the tubular neighborhood but depends on the isotopy class of knot . In particular, we defined the wrapped Floer cohomology as an invariant of the knot .We denote the resulting common graded group by
which is called the knot Floer algebra in [BKO]. Since the group is independent of the choice of tubular neighborhood of , one may regard this group as the Moore homology version (in the horizontal direction) of the wrapped Floer cohomology of the asymptotic boundary of non-compact manifold . (See [BKO].)
1.1. Formality of Floer complex
In this paper, we specialize our focus on the case of hyperbolic knots, i.e., of the knots (or links) such that the complement admits a complete metric of constant curvature . We exploit the presence of hyperbolic metric on the complement for the computation of , even though the metric cannot be smoothly extended to itself. In other words, the wrapping we put in the definition of wrapped Floer cohomology is of different nature from that of [BKO]. However, it is an interesting symplectic topological property of hyperbolic knots (or links) that is convex at in the sense that it admits a -pluri-subharmonic exhaustion function (Proposition 4.2 for the Sasakian almost complex structure associated to the hyperbolic metric , while as already mentioned in the introduction of [BKO], it may not be convex for general knots. The precise construction will be given in Section 3. We utilize the special geometric property in the calculation of the associated structures, by considering () associated to a special choice of the above mentioned tubular neighborhood so that each component of whose boundary is given by a horo-torus contained in the cusp-neighborhood of with respect to the hyperbolic metric. Although we will mostly restrict ourselves to the case of knots for the simplicity of exposition, we would like to emphasize that main results of the present paper also apply to the links whose ramification to the study of links is worthwhile to investigate.
Consider the kinetic energy Hamiltonian of the hyperbolic metric on . We first prove the following general properties of the Hamiltonian chords associated to the conormal and their associated geodesic cords attached to . 111We will follow the terminology adopted by Ng [Ng] the term chord for the Hamiltonian trajectory attached to the conormal and the term cord for the corresponding geodesic attached to the base of the conormal.
Theorem 1.1 (Theorem 6.3).
Let and be as above. Then for any geodesic cord , both Morse index and nullity of vanish. In particular, any non-constant Hamiltonian chord associated to is rigid and nondegenerate.
This enables us to work with the kinetic energy Hamiltonian, without perturbation, in the construction of an algebra generated by the set of Hamiltonian chords of attached to . The relevant perturbed pseudo-holomorphic equation is nothing but
for a map satisfying suitable (moving) Lagrangian boundary condition together with asymptotic conditions converging to Hamiltonian chords of the kinetic energy Hamiltonian given above. We especially study the Cauchy-Riemann equation (1.1) with respect to the Sasakian almost complex structure on of on : It is given by
under the splitting via the Levi-Civita connection of .
The graded Floer chain complex is a free abelian group generated by the Hamiltonian chords:
where is a cochain complex of , e.g., the de Rham complex of or the Morse complex of a Morse function of . Here the grading is given by the grading of the Hamiltonian chords . We establish the estimates, especially the horizontal estimate, in Section 8.2, which enables us to directly define the wrapped Floer complex as an algebra for the hyperbolic metric, without making a cylindrical adjustment unlike in [BKO].
We show that forms a sub-complex of and so can define the reduced complex
and denote its induced operators by and its cohomology by .
The following formality of is the first main theorem we prove in the present paper.
Theorem 1.2 (Theorem 9.1).
Let be the hyperbolic metric of and be the Sasakian almost complex structure of on , and let be a horo-torus as above. Consider the kinetic energy Hamiltonian associated to the metric
and the associated perturbed Cauchy-Riemann equations (1.1) equipped with some boundary condition associated to the conormal . Let be the corresponding maps. Then we have for all .
This theorem is a consequence of the standard Fredholm theory combined with the geometric properties of the hyperbolic metric stated in Theorem 1.1 in the study of moduli space of (1.1). (See Section 6 for details.)
The next theorem is the first step towards making an explicit calculation of the map . To describe the matrix coefficients of , we will relate solutions of (1.1) to the hyperbolic triangles on truncated by . We recall that the projection of non-constant Hamiltonian trajectory of is a geodesic cord and conversely each geodesic is uniquely lifted to a Hamiltonian chord.
Let and be as above and be any solution to (1.1) with its asymptotic triple of Hamiltonian chords. Denote by the associated geodesic cords for the pair . For each solution of (1.1) in , we lift it to the universal covering space together with the relevant boundary and asymptotic conditions. Then the triple of geodesic cords determines the triple of points in modulo the action of deck transformations. (See Lemma 11.2.) We denote by the canonical projection.
the totally geodesic triangle in determined by and let as the unique lift of satisfying the asymptotic conditions given by the triple. Then the map defined by has its image contained in .
The statements in Theorem 1.3 can be written in terms of as follows.
Let be any solution to (1.1) with its asymptotic triple of Hamiltonian chords. Denote by the associated geodesics on . There exists a totally geodesic immersed ideal triangle whose ideal edges contain each geodesic triples such that the map defined by
has its image contained in .
The following conjecture is an important one to resolve.
The map induces a one-one correspondence between the set of Floer triangles associated to the triple and that of geodesic triangle associated to the triple with .
One outcome of Conjecture 1.5 would be that calculation of structure constants of the algebra is reduced to a counting problem of geodesic triangles. We hope to come back to investigate validity of this conjecture elsewhere.
1.2. Comparison of with Knot Floer Algebra
Finally we prove a comparison result between with the Knot Floer Algebra introduced in [BKO] which is described in the beginning of this introduction. Combining some known result, Proposition 3.6, on hyperbolic geometry of relative to its ideal boundary, we obtain
Theorem 1.6 (Theorem 10.3).
Suppose is a hyperbolic knot on . Then we have an (algebra) isomorphism
for all integer . In particular for all and is a free abelian group generated by .
We refer to Definition 3.5 for the set of infinite tame geodesics.
It is worthwhile to state the contrapositive of the second statement separately.
Let be a knot such that for some . Then the knot cannot be hyperbolic.
We emphasize that in the context of hyperbolic knots which is the case of our main interest in the present paper the given hyperbolic metric on is neither cylindrical at infinity nor smoothly extends to the whole space . Because of this, we cannot directly use the hyperbolic metric defined on for the calculation of . The essence of the proof of Theorem 1.6 is to establish the existence of an isomorphism
for a cylindrical adjustment of induced by the homotopy . Since the two metrics and are not Lipschitz equivalent, the result from [BKO] does not apply to this pair but should be proved separately. The proof of (1.3) consists of three essential steps in order:
The first step is to establish the existence of the continuation map
For this purpose, we take the cylindrical adjustment so that and establish a -estimate for the relevant non-autonomous perturbed Cauchy-Riemann equation. (See Proposition 9.3 and its proof.)
Construct a direct system and establish the isomorphism
for a sequence of cylindrical adjustments with . The proof strongly relies on the formality of the complex (and of which relies on the fact that the metric also can be chosen so that it has non-positive curvature everywhere. See Proposition 9.6.)
The final step is to prove that the natural map
is an isomorphism. This step also relies on the formality of the associated complexes.
1.3. Hyperbolic geometry and the Bochner techniques
The main ingredients of the proofs of both theorems mentioned above are various applications of the Bochner-type techniques both in the pointwise version and in the integral version. A brief outline of how we apply these techniques is now in order.
Theorem 1.2 is a consequence of Theorem 1.1 and a degree counting argument. Therefore we will focus on Theorem 1.1 and Theorem 1.3. The proof of Theorem 1.1 is an explicit calculation of the second variation of the energy functional for the paths satisfying the free boundary condition associated to the horo-torus . Then the explicit formula we obtain manifestly establishes the positivity and nondegeneracy of the second variation, which is thanks to the special geometric property of the horo-torus (See Section 6.): it has a constant positive mean curvature relative to the outward unit normal to pointing to the cusp direction, which we also compute.
For the proof of Theorem 1.3, we apply some isometry element , and first reduce the classification problem to that of .
For these purposes, we exploit
the negative constant curvature property of hyperbolic metric on ,
the constant mean-curvature property of the horo-torus with the correct sign,
a usage of (strong) maximum principle based on rather delicate calculation of the Laplacian of an indicator function and subtle rearrangement of the terms appearing in the computed Laplacian. (See Appendix A.)
In the literature on symplectic geometry, Hamiltonian dynamics, contact geometry and the physics literature, there are various conventions used which are different from one another one way or the other. In the mathematics literature, there are two conventions that have been dominantly appeared, which are summarized in the preface of the book [Oh4]: one is the convention that has been consistently used by the third named author and the other is the one that is called Entov-Polterovich’s convention in [Oh4].
The major differences between the two conventions lie in the choice of the following three definitions:
Definition of Hamiltonian vector field: On a symplectic manifold , the Hamiltonian vector field associated to a function is given by the formula
Compatible almost complex structure: In both conventions, is compatible to if the bilinear form is positive definite.
Canonical symplectic form: On the cotangent bundle , the canonical symplectic form is given by
It appears that in the physics literature (e.g., [AENV] and others) the canonical symplectic form is taken as as well as in [Kl, EENS] and [Oh2]-[Oh5]. For the convenience of designating the conventions, let us call the first Convention I and the second Convention II in the paragraph below. Our current convention is consistent with that of [FOOO].
We will utilize various forms of the (strong) maximum principle for the equation
with the negative sign in front of . In the strip coordinate , the equation becomes
Applicability of the maximum principle is very sensitive to the choice of conventions and the signs in the relevant equations in general such as (1.1) in the present study. We study the Cauchy-Riemann equation (1.1) on associated to the triple
where , and are defined on following Convention I. Under these circumstances, it turns out that it is essential to adopt Convention I to be able to apply the various maximum principles we need for the equation (1.1). (See the calculations provided in Appendix and Subsection 8.2, especially Lemma 8.3, to see how these arise.)
In addition to these, the Floer continuation map is defined over the homotopy of Hamiltonian in the increasing direction. To be able to obtain the necessary energy estimates in the wrapped setting, we consider the action functional associated to Hamiltonian on by
which is the negative of the classical action functional. For the kinetic energy Hamiltonian , we have
where is the Hamiltonian chord associated to the geodesic and is the energy of with respect to the metric .
While this paper is written as a sequel to [BKO], its content is largely independent of that of [BKO] except that we adopt the same convention as thereof. Except in Section 2 and 10, we directly work with the given hyperbolic metric for the study of perturbed Cauchy-Riemann equation above without taking the cylindrical adjustment. This forces us to establish a new form of horizontal estimates (see Theorem 8.4) directly applicable to the hyperbolic metric without taking a cylindrical adjustment. One important difference of the hyperbolic metric from that of cylindrical metric is that a geodesic issued even inward from for may go out of the domain , get closer to the knot and then come back to the domain . In Section 10, we compare the wrapped Floer cohomology associated to with the Knot Floer cohomology defined for a general knot, not necessarily a hyperbolic knot, via a cylindrical adjustment of a smooth metric on restricted to .
Acknowledgement: Y. Bae thanks Research Institute for Mathematical Sciences, Kyoto University for its warm hospitality.
2. Definition of Knot Floer algebra in [Bko]
We first provide the construction of Knot Floer algebra introduced in [BKO] without the details of its construction.
Let be a smooth Riemannian metric on . Consider a tubular neighborhood of . We denote its boundary by and , the conormal bundle of the torus .
We define a cylindrical adjustment of the metric on by
which is suitably interpolated on and fixed. Then we denote
We denote by the set of Hamiltonian chords of attached to a Lagrangian submanifold in general. We have
where the subindex of in the right hand side denotes the action of the Hamiltonian chords of . We also define
and call the action spectrum of the pair . By definition of the kinetic energy Hamiltonian
We note that and the component is clean in the sense of Bott as follows.
For any metric , the set of constant Hamiltonian cords of is the union
where is the set of constant paths valued at a point in whose domain is regarded as . Each is in one-one correspondence with .
Each is normally nondegenerate in the path space
and diffeomorphic to , provided for .
For a generic choice of metric , all non-constant Hamiltonian cords are non-degenerate and their associated geodesics have lengths not equal to , or equivalently
Statement (3) is standard and so omitted. (See [Ab] and the discussion and the proofs in [Oh3, Section 3] for the relevant Fredholm setting for such a proof in the more complicated non-exact context.) The statement (1) is also a direct consequence of the boundary condition , since any constant solution of has zero momentum, i.e., .
The remaining proof will be occupied by the proof of Statement (2). Let with be a constant Hamiltonian cord valued at .
For each pair of vector fields along satisfying
a straightforward calculation give rise to the formula for the Hessian of at
(We refer readers to the proof of [Oh5, Proposition 18.2.8] for the calculation of the first term in the first line, the Hessian of the functional .) Therefore a kernel element of is given by the vector field satisfying
This is a first order ODE with constant coefficient matrix on the vector space . Therefore the general solution of this ODE is given by
for each . Furthermore the final condition implies
On the other hand, we have
where is the identity map on . Therefore we have
By decomposing into the components of and under the decomposition , we rewrite the equation
Then using the property of Sasakian almost complex structure , we obtain
Finally we aplpy
From this, we derive since provided . Therefore we have derived , i.e.,
Conversely, we can check that any constant vector field satisfies the above kernel equation. This proves which finishes the proof. ∎
where is a cochain complex of , e.g., the de Rham complex and associate an algebra following the construction from [FOOO]. It was shown in [BKO] that does not depend on the choice of smooth metric on and of the tubular neighborhood but depends only on the isotopy type of the knot .
Definition 2.2 (Knot Floer algebra [Bko]).
We denote by
the resulting common (isomorphism class of the) group and call it the knot Floer algebra of in .
To facilitate our calculation of this algebra and its comparison with Knot Floer algebra , we now take the Morse complex model for , and realize the model (2.2) as a nondegenerate wrapped Floer complex of a perturbed as follows.
Take a compactly supported smooth function satisfying in a neighborhood of , and consider the translated conormal whose fiber is given by
Then it is easy to check
and the intersection is nondegenerate if and only if .
We then take a radially cut-off function satisfying where is a monotonically increasing function satisfying
and consider the function defined by
Then we take a Darboux-Weinstein chart of and then consider the exact Lagrangian submanifold
In other words, for the Lagrangian embedding defined by
As usual, we require to satisfy
under the canonical identifications of and
It is easy to check that satisfies and so is an exact Lagrangian submanifold. We also note that
and for and for .
Next we denote by the energy of the shortest geodesic cord of relative to the metric . Then the following lemma is an immediate consequence of the implicit function theorem.
Let be the function given above such that . Then there exists some such that
provided . Here means one-one correspondence.
Both identities then are immediate consequences of Sard-Smale implicit function theorem via the definition of the cut-off function and -smallness of , and the requirement (2.6). ∎
By the generic transversality proof under the perturbation of Lagrangian boundary from [Oh1], we can choose so that is nondegenerate. We have one-one correspondence
We denote . Then we define a Floer chain complex
Here the grading is given by the grading of the Hamiltonian chords . Then the construction in [BKO] associates an algebra to . We denote the associated cohomology by
It follows from the Bott-Morse property of that the complex has such a decomposition
Moreover is a subcomplex of which is isomorphic to the Morse complex of the Morse function .
The operator has the matrix form
with respect to the above decomposition.
Definition 2.5 (Reduced Knot Floer complex).
Denote by the quotient complex
We call this complex by the reduced Knot Floer complex.
We also note and so
is naturally isomorphic to the reduced complex .
Therefore the reduced complex is nothing but the complex generated by non-constant Hamiltonian chords.
We emphasize that in the context of hyperbolic knots which is the case of our main interest in the present paper the given hyperbolic metric on is neither cylindrical at infinity nor smoothly extends to the whole space . Because of this, we cannot directly use the hyperbolic metric defined on for the calculation of . The rest of the paper is occupied by the construction of this wrapped Floer complex associated to the kinetic energy Hamiltonian of the hyperbolic metric on whose injectivity radius is zero.
3. Preliminary on hyperbolic 3-manifold of finite volume
In this section, we briefly review several well-known facts. For a reference, Martelli’s book [M] is readable and enough to know some basics about hyperbolic geometry and 3-manifold theory used in this article.
Let be a complete hyperbolic 3-manifold. The universal cover is identified with the hyperbolic 3-space
and is isometric to with a discrete group . If is orientable, consists of orientation preserving isometries. Hence, from now on we identify the hyperbolic space and the group of orientation preserving isometries with a upper half space and respectively. The ideal boundary of is identified with and acts on and as Möbius transformations, and the Poincaré extensions, respectively.
Let be a knot complement of a orientable closed 3-manifold , i.e., . Then is homeomorphic to the interior of the knot exterior, a compact 3-manifold denoted by , that has a torus boundary of and the complete hyperbolic structure should be of finite volume by the torus boundary condition.
3.1. -thick-thin decomposition by the Busemann function
The first important fact in this situation would be the uniqueness of the hyperbolic metric by Mostow-Prasad rigidity, i.e., is unique up to isometry. Therefore any hyperbolic metric invariant can be regarded as a topological invariant, as we have a canonical Riemannian metric.
Secondly, we can nicely separate compact part and the non-compact end part of as follow.
There is a constant such that the -thin part,
is homeomorphic to the end of , which is . Thus the -thick part is compact and the interior is homeomorphic to itself once we take ,
By the thick-thin decomposition using the Margulis constant , we have two kinds of thin parts, thin-tubes and truncated cusp . When we take less than smallest injective radius among thin-tubes, then the only possible -then part is the truncated cusp of the boundary. ∎
By the structural property of the thick-thin decomposition [M, Section 4], we know that is a truncated cusp and thus is a Euclidean torus. We describe by using the Busemann function instead of injective radius.
Let be a geodesic ray satisfying
The Busemann function is defined by
Without loss of generality, we can take a lifting in of ,
such that the lifted Busemann function