Bordered Floer Homology Research Articles

Involutive knot Floer homology and bordered modules

We prove that, up to local equivalences, a suitable truncation of the involutive knot Floer homology of a knot in S^3 and the involutive bordered Heegaard Floer theory of its complement determine each other. In particular, given two knots K_1 and K_2 , we prove that the \mathbb{F}_2[U,V]/(UV) -coefficient involutive knot Floer homology of K_1 \sharp -K_2 is \iota_K -locally trivial if \widehat{CFD}(S^3 \backslash K_1) and \widehat{CFD}(S^2 \backslash K_2) satisfy a certain condition which can be seen as the bordered counterpart of \iota_K -local equivalence. We further establish an explicit algebraic formula that computes the hat-flavored truncation of the involutive knot Floer homology of a knot from the involutive bordered Floer homology of its complement. It follows that there exists an algebraic satellite operator defined on the local equivalence group of knot Floer chain complexes, which can be computed explicitly up to a suitable truncation.

Bordered Floer homology for manifolds with torus boundary via immersed curves

This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M M is such a manifold, we show that the type D structure C F D ^ ( M ) \widehat {CFD}(M) may be viewed as a set of immersed curves decorated with local systems in ∂ M \partial M . These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism h h between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of H F ^ \widehat {HF} decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of H F ^ \widehat {HF} . In particular, it follows that a prime rational homology sphere Y Y with H F ^ ( Y ) > 5 \widehat {HF}(Y)>5 must be geometric. Other results include a new proof of Eftekhary’s theorem that L-space homology spheres are atoroidal; a complete characterization of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots.

A calculus for bordered Floer homology

A calculus for bordered Floer homology

Bordered Floer homology and contact structures

Abstract We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $(M, \xi , \mathcal {F})$ whose convex boundary is equipped with a signed singular foliation $\mathcal {F}$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $c_D$ and $c_A$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $\mathcal {F}$ in favor of the dividing set and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda–Kazez–Matić.

A friendly introduction to the bordered contact invariant

We give a short introduction to the contact invariant in bordered Floer homology defined by F\"oldv\'ari, Hendricks, and the authors. The construction relies on a special class of foliated open books. We discuss a procedure to obtain such a foliated open book and present a definition of the contact invariant. We also provide a "local proof", through an explicit bordered computation, of the vanishing of the contact invariant for overtwisted structures.

Bordered Floer homology and a meridional class of knot

For a knot K and its knot Floer complex \(CFK^-(K)\), we introduce an algorithm to compute the bordered Floer bimodule of the complement of the knot and its meridian. The grading of the module computes \(spin^c\)-summands of \({\widehat{HFK}}(S^3_{-n}(K), \mu _K)\), which can also be extended to arbitrary framing n.

Singular crossings and Ozsváth–Szabó’s Kauffman-states functor

Recently, Ozsváth and Szabó introduced some algebraic constructions computing knot Floer homology in the spirit of bordered Floer homology, including a family of algebras $\mathcal B (n)$ and, for a generator of the braid group on $n$ strands, a certain t

Strands algebras and Ozsváth and Szabó’s Kauffman-states functor

We define new differential graded algebras A(n,k,S) in the framework of Lipshitz-Ozsv\'ath-Thurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsv\'ath-Szab\'o's algebras B(n,k,S); indeed, we exhibit a quasi-isomorphism from B(n,k,S) to A(n,k,S). We also show how Ozsv\'ath-Szab\'o's gradings on B(n,k,S) arise naturally from the general framework of group-valued gradings on strands algebras.

Cornered Heegaard Floer Homology

Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3-manifolds with codimension-2 corners, and prove that the bordered Floer homology of a 3-manifold with boundary, split into two pieces with corners, can be recovered as a tensor product of the cornered invariants of the pieces.

Involutive bordered Floer homology

We give a bordered extension of involutive HF-hat and use it to give an algorithm to compute involutive HF-hat for general 3-manifolds. We also explain how the mapping class group action on HF-hat can be computed using bordered Floer homology. As applications, we prove that involutive HF-hat satisfies a surgery exact triangle and compute HFI-hat of the branched double covers of all 10-crossing knots.

Strand algebras and contact categories

We demonstrate an isomorphism between the homology of the strand algebra of bordered Floer homology, and the category algebra of the contact category introduced by Honda. This isomorphism provides a direct correspondence between various notions of Floer homology and arc diagrams, on the one hand, and contact geometry and topology on the other. In particular, arc diagrams correspond to quadrangulated surfaces, idempotents correspond to certain basic dividing sets, strand diagrams correspond to contact structures, and multiplication of strand diagrams corresponds to stacking of contact structures. The contact structures considered are cubulated, and the cubes are shown to behave equivalently to local fragments of strand diagrams.

Relative Q-Gradings from Bordered Floer Theory

In this paper we show how to recover the relative Q-grading in Heegaard Floer homology from the noncommutative grading on bordered Floer homology.

Bordered Heegaard Floer homology

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A-infinity module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A-infinity tensor product of the type D module of one piece and the type A module from the other piece is HF^ of the glued manifold. As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for HF^. We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.

Bordered Floer homology and existence of incompressible tori in homology spheres

Let$Y$be a homology sphere which contains an incompressible torus. We show that$Y$cannot be an$L$-space, i.e. the rank of$\widehat{\text{HF}}(Y)$is greater than$1$. In fact, if the homology sphere$Y$is an irreducible$L$-space, then$Y$is$S^{3}$, the Poincaré sphere$\unicode[STIX]{x1D6F4}(2,3,5)$or hyperbolic.

The decategorification of bordered Heegaard Floer homology

Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism \phi from F to \bdy Y, a module over A(Z). We study the Grothendieck group of modules over A(Z), and define an invariant lying in this group for every bordered 3-manifold. We prove that this invariant recovers the kernel of the inclusion of H_1(\bdy Y; Z) into H_1(Y; Z) if H_1(Y, \bdy Y; Z) is finite, and is 0 otherwise. We also study the properties of this invariant corresponding to gluing. As one application, we show that the pairing theorem for bordered Floer homology categorifies the classical Alexander polynomial formula for satellites.

Bordered Floer homology of (2,2n)-torus link complement

We compute bordered Floer homology CFDD of (2,2n)-torus link complement, and discuss assorted examples and type-DD structure homotopy equivalence.

The Alexander module, Seifert forms, and categorification

We show that bordered Floer homology provides a categorification of a TQFT described by Donaldson. This, in turn, leads to a proof that both the Alexander module of a knot and the Seifert form are completely determined by Heegaard Floer theory.

Combinatorial tangle Floer homology

In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$ and $I\times S^2$. The special case of $S^3$ gives back a stabilized version of knot Floer homology.

A type A structure in Khovanov homology

Inspired by bordered Floer homology, we describe a type A structure on a Khovanov homology for a tangle, which complements the type D structure in a previous paper. The type A structure is a differential module over a certain algebra. This can be paired with the type D structure to recover the Khovanov chain complex. The homotopy type of the type A structure is a tangle invariant, and homotopy equivalences of the type A structure result in chain homotopy equivalences on the Khovanov chain complex. We can use this to simplify computations and introduce a modular approach to the computation of Khovanov homologies. This approach adds to the literature even in the case of a connect sum, where the techniques here will allow an exact computation of Khovanov homology from the structures for two tangles coming from the summands. Several examples are included, showing in particular how we can compute the correct torsion summands for the Khovanov homology of the connect sum. A lengthy appendix is devoted to establishing the theory of these structures over a characterstic zero ring.

Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree

Given a link in the three-sphere, Ozsv\'ath and Szab\'o showed that there is a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double cover. The aim of this paper is to explicitly calculate this spectral sequence in terms of bordered Floer homology. There are two primary ingredients in this computation: an explicit calculation of bimodules associated to Dehn twists, and a general pairing theorem for polygons. The previous part (arXiv:1011.0499) focuses on computing the bimodules; this part focuses on the pairing theorem for polygons, in order to prove that the spectral sequence constructed in the previous part agrees with the one constructed by Ozsv\'ath and Szab\'o.