Floer Homology Research Articles

A computation of the ring structure in wrapped Floer homology

We give an explicit computation of the ring structure in wrapped Floer homology of a class of real Lagrangians in \(A_k\)-type Milnor fibers. In the \(A_k\)-type plumbing description, those Lagrangians correspond to the cotangent fibers or the diagonal Lagrangians. The main ingredient of the computation is to apply a version of the Seidel representation. For a technical reason, we first carry out computations in v-shaped wrapped Floer homology, and this in turn gives the desired ring structure via the Viterbo transfer map.

More concordance homomorphisms from knot Floer homology

We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring $\mathbb{F}[U, V]/(UV=0)$. We compare our invariants to other concordance homomorphisms coming from knot Floer homology, and discuss applications to topologically slice knots, concordance genus, and concordance unknotting number.

Instantons and Bar-Natan homology

A spectral sequence is established whose $E_{2}$ page is Bar-Natan's variant of Khovanov homology and which abuts to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence whose $E_{2}$ page is a characteristic-2 version of $F_{5}$ homology in Khovanov's classification.

Link Floer homology also detects split links

Inspired by work of Lipshitz-Sarkar, we show that the module structure on link Floer homology detects split links. Using results of Ni, Alishahi-Lipshitz, and Lipshitz-Sarkar, we establish an analogous detection result for sutured Floer homology.

On cobordism maps on periodic Floer homology

In this article, we investigate the cobordism maps on periodic Floer homology (PFH). In the first part of the paper, we define the cobordism maps on PFH via Seiberg Witten theory as well as the isomorphism between PFH and Seiberg Witten cohomology. Furthermore, we show that the maps satisfy the holomorphic curve axiom. In the second part of the paper, we give an alternative definition of these maps by using holomorphic curve method, provided that the symplectic cobordisms are Lefschetz fibration satisfying certain nice conditions. Under additionally certain monotonicity assumptions, we show that these two definitions are equivalent.

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.

0-Concordance of 2-knots

In this paper we investigate the 0-concordance classes of 2-knots in S 4 S^4 , an equivalence relation that is related to understanding smooth structures on 4-manifolds. Using Rochlin’s invariant, and invariants arising from Heegaard–Floer homology, we will prove that there are infinitely many 0-concordance classes of 2-knots.

Equivariant wrapped Floer homology and symmetric periodic Reeb orbits

AbstractThe aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex star-shaped hypersurfaces in ${\mathbb {R}}^{2n}$ which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting.

A refinement of the Ozsváth-Szabó large integer surgery formula and knot concordance

We compute the knot Floer filtration induced by a cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. We give a formula in terms of the original knot Floer complex of the knot in the three-sphere. As an application, we show that a knot concordance invariant of Hom can equivalently be defined in terms of filtered maps on the Heegaard Floer homology groups induced by the two-handle attachment cobordism of surgery along a knot.

Khovanov homology detects the figure‐eight knot

Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology, we prove that reduced Khovanov homology (over $\mathbb{Q}$) detects the figure-eight knot.

Instantons and some concordance invariants of knots

Concordance invariants of knots are derived from the instanton homology groups with local coefficients, as introduced in earlier work of the authors. These concordance invariants include a 1-parameter family of homomorphisms $f_{r}$, from the knot concordance group to the reals. Prima facie, these concordance invariants have the potential to provide independent bounds on the genus and number of double points for immersed surfaces with boundary a given knot.

A note on the weak splitting number

The weak splitting number wsp ⁡ ( L ) \operatorname {wsp}(L) of a link L L is the minimal number of crossing changes needed to turn L L into a split union of knots. We describe conditions under which certain R \mathbb {R} -valued link invariants give lower bounds on wsp ⁡ ( L ) \operatorname {wsp}(L) . This result is used both to obtain new bounds on wsp ⁡ ( L ) \operatorname {wsp}(L) in terms of the multivariable signature and to recover known lower bounds in terms of the τ \tau and s s -invariants. We also establish new obstructions using link Floer homology and apply all these methods to compute wsp \operatorname {wsp} for all but two of the 130 130 prime links with nine or fewer crossings.

Note on coisotropic Floer homology and leafwise fixed points

<p style='text-indent:20px;'>For an adiscal or monotone regular coisotropic submanifold <inline-formula><tex-math id="M1">$ N $</tex-math></inline-formula> of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of <inline-formula><tex-math id="M2">$ N $</tex-math></inline-formula>. Given a Hamiltonian isotopy <inline-formula><tex-math id="M3">$ \varphi = ( \varphi^t) $</tex-math></inline-formula> and a suitable almost complex structure, the corresponding Floer chain complex is generated by the <inline-formula><tex-math id="M4">$ (N, \varphi) $</tex-math></inline-formula>-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold.</p><p style='text-indent:20px;'>Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine.</p><p style='text-indent:20px;'>The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.</p>

Boundary $\mathcal{N} = 2$ theory, Floer homologies, affine algebras, and the Verlinde formula

Generalizing our ideas in [arXiv:1006.3313], we explain how topologically-twisted N=2 gauge theory on a four-manifold with boundary, will allow us to furnish purely physical proofs of (i) the Atiyah-Floer conjecture, (ii) Munoz's theorem relating quantum and instanton Floer cohomology, (iii) their monopole counterparts, and (iv) their higher rank generalizations. In the case where the boundary is a Seifert manifold, one can also relate its instanton Floer homology to modules of an affine algebra via a 2d A-model with target the based loop group. As an offshoot, we will be able to demonstrate an action of the affine algebra on the quantum cohomology of the moduli space of flat connections on a Riemann surface, as well as derive the Verlinde formula.

Ribbon knots, cabling, and handle decompositions

The fusion number of a ribbon knot is the minimal number of 1-handles needed to construct a ribbon disk. The strong homotopy fusion number of a ribbon knot is the minimal number of 2-handles in a handle decomposition of a ribbon disk complement. We demonstrate that these invariants behave completely differently under cabling by showing that the (p,1)-cable of any ribbon knot with fusion number one has strong homotopy fusion number one and fusion number p. Our main tools are Juhasz-Miller-Zemke's bound on fusion number coming from the torsion order of knot Floer homology and Hanselman-Watson's cabling formula for immersed curves.

Knot homologies in monopole and instanton theories via sutures

In this paper we construct possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot $K\subset Y$ and a base point $p\in K$, we can associate the minus versions, $\underline{\rm KHM}^-(Y,K,p)$ and $\underline{\rm KHI}^-(Y,K,p)$, to the triple $(Y,K,p)$. We prove that a Seifert surface of $K$ induces a $\mathbb{Z}$-grading, and there is an $U$-map on the minus versions, which is of degree $-1$. We also prove other basic properties of them. If $K\subset Y$ is not null-homologous but represents a torsion class, then we can also construct the corresponding minus versions for $(Y,K,p)$. We also proved a surgery-type formula relating the minus versions of a knot $K$ with those of the dual knot, when performing a Dehn surgery of large enough slope along $K$. The techniques developed in this paper can also be applied to compute the sutured monopole and instanton Floer homologies of any sutured solid tori.

Knot Floer homology and strongly homotopy-ribbon concordances

We prove that the map on knot Floer homology induced by a strongly homotopy-ribbon concordance is injective.

The shift map on Floer trajectory spaces

In this article we give a uniform proof why the shift map on Floer homology trajectory spaces is scale smooth. This proof works for various Floer homologies, periodic, Lagrangian, Hyperk\"ahler, elliptic or parabolic, and uses Hilbert space valued Sobolev theory.

Heegaard Floer homology and splicing homology spheres

We prove a basic inequality for the d-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving results of the first and second authors. As a corollary, a degree one map between two aspherical Seifert homology spheres is homotopic to a homeomorphism if and only if the Heegaard Floer homologies are isomorphic.

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