Sutured Floer Homology Research Articles

Split link detection for sl(P)$\mathfrak {sl}(P)$ link homology in characteristic P$P$

AbstractWe provide a sufficient condition for splitness of a link in terms of its reduced link homology with arbitrary field coefficients. The proof of sufficiency uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients. If is prime and the coefficient field is of characteristic , then the sufficient condition for splitness is also necessary. When , we recover Lipshitz–Sarkar's split link detection result for Khovanov homology with coefficients.

On Contact Surgery and Knot Floer Invariants

Abstract We establish some general relations between Heegaard-Floer-based contact invariants. In particular, we observe that if the contact invariant of large negative, respectively positive, contact surgeries along a Legendrian knot does not vanish, then the Legendrian invariant, respectively the Legendrian inverse limit invariant, of that knot is non-zero. We use sutured Floer homology and the limit constructions due to Golla and Etnyre, Vela-Vick and Zarev.

On Floer minimal knots in sutured manifolds

Suppose ( M , γ ) (M, \gamma ) is a balanced sutured manifold and K K is a rationally null-homologous knot in M M . It is known that the rank of the sutured Floer homology of M ∖ N ( K ) M\backslash N(K) is at least twice the rank of the sutured Floer homology of M M . This paper studies the properties of K K when the equality is achieved for instanton homology. As an application, we show that if L ⊂ S 3 L\subset S^3 is a fixed link and K K is a knot in the complement of L L , then the instanton link Floer homology of L ∪ K L\cup K achieves the minimum rank if and only if K K is the unknot in S 3 ∖ L S^3\backslash L .

Concordance maps in HFK−

We show that a decorated knot concordance [Formula: see text] from [Formula: see text] to [Formula: see text] induces an [Formula: see text]-module homomorphism [Formula: see text] which preserves the Alexander and absolute [Formula: see text]-Maslov gradings. Our construction generalizes the concordance maps induced on [Formula: see text] studied by Juhász and Marengon [Concordance maps in knot Floer homology, Geom. Topol. 20 (2016) 3623–3673], but uses the description of [Formula: see text] as a direct limit of maps between sutured Floer homology groups discovered by Etnyre et al. [Sutured Floer homology and invariants of Legendrian and transverse knots, Geom. Topol. 21 (2017) 1469–1582].

Concordance surgery and the Ozsváth–Szabó 4-manifold invariant

We compute the effect of concordance surgery, a generalization of knot surgery defined using a self-concordance of a knot, on the Ozsváth–Szabó 4-manifold invariant. The formula involves the graded Lefschetz number of the concordance map on knot Floer homology. The proof uses the sutured Floer TQFT, and a version of sutured Floer homology perturbed by a 2-form.

Rank inequalities for the Heegaard Floer homology of branched covers

Given a double cover between 3-manifolds branched along a nullhomologous link, we establish an inequality between the dimensions of their Heegaard Floer homologies. We discuss the relationship with the L-space conjecture and give some other topological applications, as well as an analogous result for sutured Floer homology.

Transverse invariants and exotic surfaces in the 4–ball

Using 1-twist rim surgery, we construct infinitely many smoothly embedded, orientable surfaces in the 4-ball bounding a knot in the 3-sphere that are pairwise topologically isotopic, but not ambient diffeomorphic. We distinguish the surfaces using the maps they induce on perturbed sutured Floer homology. Along the way, we show that the cobordism map induced by an ascending surface in a Weinstein cobordism preserves the transverse invariant in knot Floer homology.

Naturality and Mapping Class Groups in Heegaard Floer Homology

We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the “space of Heegaard diagrams,” and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds.

On the equivalence of contact invariants in sutured Floer homology theories

We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka's sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology which identifies our invariant with the contact class defined by Honda, Kazez and Mati\'c in the latter theory. One consequence is that the Legendrian invariants in knot Floer homology behave functorially with respect to Lagrangian concordance. In particular, these invariants provide computable and effective obstructions to the existence of such concordances. Our work also provides the first proof which does not rely on the relative Giroux correspondence that the vanishing or non-vanishing of Honda, Kazez and Mati\'c's contact class is a well-defined invariant of contact manifolds.

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.

Tangle Floer homology and cobordisms between tangles

We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one-to-one correspondence with the sutured manifolds. We define cobordisms between sutured manifolds (tangles) by generalizing cobordisms between oriented tangles. For every commutative algebra A over Z/2Z, we define A-Tangles to be the category consisting of A-tangles, which are balanced tangles with A-colorings of the tangle strands and fixed SpinC structures, and A-cobordisms as morphisms. An A-cobordism is a cobordism with a compatible A-coloring and an affine set of SpinC structures. Associated with every A-module M we construct a functor $HF^M$ from A-Tangles to A-Modules, called the tangle Floer homology functor, where A-Modules denotes the the category of A-modules and A-homomorphisms between them. Moreover, for any A-tangle T the A-module $HF^M(T)$ is the extension of sutured Floer homology defined in an earlier work of the authors. In particular, this construction generalizes the 4-manifold invariants of Ozsvath and Szabo. Moreover, applying the above machinery to decorated cobordisms between links, we get functorial maps on link Floer homology.

Contact handles, duality, and sutured Floer homology

We give an explicit construction of the Honda--Kazez--Mati\'c gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link Floer homology defined by the first author via sutured manifold cobordisms and by the second author via elementary cobordisms agree.

Tight contact structures on Seifert surface complements

We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured Floer homology. Finally, we relate our methods and results to Kauffman's formal knot theory.

Sutured Floer homology and invariants of Legendrian and transverse knots

Using contact-geometric techniques and sutured Floer homology, we present an alternate formulation of the minus and plus version of knot Floer homology. We further show how natural constructions in the realm of contact geometry give rise to much of the formal structure relating the various versions of Heegaard Floer homology. In addition, to a Legendrian or transverse knot K in a contact manifold (Y,\xi), we associate distinguished classes EHL(K) in the minus-version of knot floer homology and EHIL(K) in the plus version, which are each invariant under Legendrian or transverse isotopies of K. The distinguished class EHL is shown to agree with the Legendrian/transverse invariant defined by Lisca, Ozsvath, Stipsicz, and Szabo despite a strikingly dissimilar definition. While our definitions and constructions only involve sutured Floer homology and contact geometry, the identification of our invariants with known invariants uses bordered sutured Floer homology to make explicit computations of maps between sutured Floer homology groups.

Cobordisms of sutured manifolds and the functoriality of link Floer homology

It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured Floer homology. This map is a common generalization of the hat version of the closed 3-manifold cobordism map in Heegaard Floer theory, and the contact gluing map defined by Honda, Kazez, and Matić. We show that sutured Floer homology, together with the above cobordism maps, forms a type of TQFT in the sense of Atiyah. Applied to the sutured manifold cobordism complementary to a decorated link cobordism, our theory gives rise to the desired map on link Floer homology. Hence, link Floer homology is a categorification of the multi-variable Alexander polynomial. We outline an alternative definition of the contact gluing map using only the contact element and handle maps. Finally, we show that a Weinstein sutured manifold cobordism preserves the contact element.

Sutured Floer homology, fibrations, and taut depth one foliations

For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology ($SFH$) can be used to determine all fibered classes in $H^1(M)$. Furthermore, we show that the $SFH$ of a balanced sutured manifold $(M,\gamma)$ detects which classes in $H^1(M)$ admit a taut depth one foliation such that the only compact leaves are the components of $R(\gamma)$. The latter had been proved earlier by the first author under the extra assumption that $H_2(M)=0$. The main technical result is that we can obtain an extremal $\text{Spin}^c$-structure $\mathfrak{s}$ (i.e., one that is in a `corner' of the support of $SFH$) via a nice and taut sutured manifold decomposition even when $H_2(M) \neq 0$, assuming the corresponding group $SFH(M,\gamma,\mathfrak{s})$ has non-trivial Euler characteristic.

Comparing invariants of Legendrian knots

We prove the equivalence of the invariants EH (L) and \mathcal L^–(\pm L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vértesi. In the course of the proof we relate the sutured Floer homology groups associated with a knot complement S^3\setminus K and the knot Floer homology of (S^3,K) and define intermediate Legendrian invariants.

Dimensionally reduced sutured Floer homology as a string homology

We show that the sutured Floer homology of a sutured 3-manifold of the form $(D^2 \times S^1, F \times S^1)$ can be expressed as the homology of a string-type complex, generated by certain sets of curves on $(D^2, F)$ and with a differential given by resolving crossings. We also give some generalisations of this isomorphism, computing "hat" and "infinity" versions of this string homology. In addition to giving interesting elementary facts about the algebra of curves on surfaces, these isomorphisms are inspired by, and establish further, connections between invariants from Floer homology and string topology.

Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic

In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of the sutured manifolds obtained by cutting the knot complements along the Seifert surfaces. Our examples provide the first use of sutured Floer homology, and not merely its Euler characteristic (a classical torsion), to distinguish Seifert surfaces. Our technique uses a version of Floer homology, called “longitude Floer homology” in a way that enables us to bypass the computations related to the SFH of the complement of a Seifert surface.

Twisty itsy bitsy topological field theory

We extend the topological field theory (itsy bitsy topological field theory) of our previous work from mod 2 to twisted coefficients. This topological field theory is derived from sutured Floer homology (SFH) but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has information-theoretic (itsy) and quantum-field-theoretic (bitsy) aspects. In the process we extend some results of SFH, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.