Floer Homology Research Articles

Knot Floer homology and surgery on equivariant knots

AbstractGiven an equivariant knot of order 2, we study the induced action of the symmetry on the knot Floer homology. We relate this action with the induced action of the symmetry on the Heegaard Floer homology of large surgeries on . This surgery formula can be thought of as an equivariant analog of the involutive large surgery formula proved by Hendricks and Manolescu. As a consequence, we obtain that for certain double branched covers of and corks, the induced action of the involution on Heegaard Floer homology can be identified with an action on the knot Floer homology. As an application, we calculate equivariant correction terms which are invariants of a generalized version of the spin rational homology cobordism group, and define two knot concordance invariants. We also compute the action of the symmetry on the knot Floer complex of for several equivariant knots.

Higher-dimensional Heegaard Floer homology and Hecke algebras

Given a closed oriented surface \Sigma of genus greater than 0, we construct a map \mathcal{F} from the wrapped higher-dimensional Heegaard Floer homology of the cotangent fibers of T^{*}\Sigma to the Hecke algebra associated to \Sigma and show that \mathcal{F} is an isomorphism of algebras. We also establish analogous results for punctured surfaces.

Tangle contact homology

Knot contact homology is an ambient isotopy invariant of knots and links in [Formula: see text]. The purpose of this paper is to extend this definition to an ambient isotopy invariant of tangles and prove that gluing of tangles gives a gluing formula for knot contact homology. As a consequence of the gluing formula we obtain that the tangle contact homology detects triviality of tangles.

Applications of higher‐dimensional Heegaard Floer homology to contact topology

AbstractThe goal of this paper is to set up the general framework of higher‐dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient condition for the Weinstein conjecture to hold. We discuss several classes of examples including those coming from analyzing a close cousin of symplectic Khovanov homology and the analog of the Plamenevskaya invariant of transverse links.

Homology concordance and knot Floer homology

Homology concordance and knot Floer homology

Stabilization distance bounds from link Floer homology

AbstractWe consider the set of connected surfaces in the 4‐ball with boundary a fixed knot in the 3‐sphere. We define the stabilization distance between two surfaces as the minimal such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most . Similarly, we consider a double‐point distance between two surfaces of the same genus that is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double‐point distance. We compute our invariants for some pairs of deform‐spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice‐disk analog of Problem 1.105 (B) from Kirby's problem list by showing the existence of non‐0‐cobordant slice disks.

Torus knot filtered embedded contact homology of the tight contact 3‐sphere

AbstractKnot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3‐sphere in terms of their presentation as open books and as Seifert fiber spaces. We provide Morse–Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg–Witten Floer theory developed by Hutchings and Taubes, and use them to compute the knot filtered embedded contact homology, for odd and positive.

Thin links and Conway spheres

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $\delta$ -grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant $\operatorname {HFT}$ and the Khovanov invariant $\widetilde {\operatorname {Kh}}$ that were developed by the authors in previous works.

Knot concordance invariants from Seiberg–Witten theory and slice genus bounds in 4-manifolds

We construct a new family of knot concordance invariants [Formula: see text], where [Formula: see text] is a prime number. Our invariants are obtained from the equivariant Seiberg–Witten–Floer cohomology, constructed by the author and Hekmati, applied to the degree [Formula: see text] cyclic cover of [Formula: see text] branched over [Formula: see text]. In the case [Formula: see text], our invariant [Formula: see text] shares many similarities with the knot Floer homology invariant [Formula: see text] defined by Hom and Wu. Our invariants [Formula: see text] give lower bounds on the genus of any smooth, properly embedded, homologically trivial surface bounding [Formula: see text] in a definite [Formula: see text]-manifold with boundary [Formula: see text].

Naturality and functoriality in involutive Heegaard Floer homology

We prove first-order naturality of involutive Heegaard Floer homology, and furthermore, construct well-defined maps on involutive Heegaard Floer homology associated to cobordisms between three-manifolds. We also prove analogous naturality and functoriality results for involutive Floer theory for knots and links. The proof relies on the doubling model for the involution, as well as several variations.

The Arnold conjecture for singular symplectic manifolds

In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of bm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b^m$$\\end{document}-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number of 1-periodic Hamiltonian orbits for b2m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b^{2m}$$\\end{document}-symplectic manifolds depending only on the topology of the manifold. Moreover, for bm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b^m$$\\end{document}-symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions.

Monopole Floer homology and invariant theta characteristics

AbstractWe describe a relationship between the monopole Floer homology of three‐manifolds and the geometry of Riemann surfaces. For an automorphism of a compact Riemann surface with quotient , there is a natural correspondence between theta characteristics on which are invariant under and self‐conjugate structures on the mapping torus of . We show that the monopole Floer homology groups of are explicitly determined by the eigenvalues of the (lift of the) action of on , the space of holomorphic sections of , and discuss several consequences of this description. Our result is based on a detailed analysis of the transversality properties of the Seiberg–Witten equations for suitable small perturbations.

New invariants for virtual knots via spanning surfaces

We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative. Finally, we use the Behrens–Golla [Formula: see text]-invariant to obstruct a knot from being a stabilization of another.

On the decategorification of some higher actions in Heegaard Floer homology

On the decategorification of some higher actions in Heegaard Floer homology

Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

Frøyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in 3- and 4-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal L -spaces. In particular, we study relations between Frøyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive explicit upper bounds for the Frøyshov invariants of minimal hyperbolic L -spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Frøyshov invariants of minimal L -spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Frøyshov invariants for all spin ^{c} structures on the Seifert–Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.

The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I

Given an open book decomposition (S,h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(S,\\mathfrak{h} )$\\end{document} adapted to a closed, oriented 3-manifold M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M$\\end{document}, we define a chain map Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Phi $\\end{document} from a certain Heegaard Floer chain complex associated to (S,h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(S,\\mathfrak{h} )$\\end{document} to a certain embedded contact homology chain complex associated to (S,h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(S,\\mathfrak{h} )$\\end{document}, as defined in (Colin et al. in Geom. Topol., 2024), and prove that it induces an isomorphism on the level of homology. This implies the isomorphism between the hat version of Heegaard Floer homology of −M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$-M$\\end{document} and the hat version of embedded contact homology of M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M$\\end{document}.

The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II

This paper is the sequel to (Colin et al. in Publ. Math. Inst. Hautes Études Sci., 2024), and is devoted to proving some of the technical parts of the HF=ECH isomorphism.

Murasugi sum and extremal knot Floer homology

The aim of this paper is to study the behavior of knot Floer homology under Murasugi sum. We establish a graded version of Ni’s isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that \tau=g for each summand if and only if \tau=g holds for the Murasugi sum (with \tau and g defined appropriately for multi-component links). Some applications are presented.

Correction to the article Bimodules in bordered Heegaard Floer homology

Correction to the article Bimodules in bordered Heegaard Floer homology

Equivariant Lagrangian Floer homology via cotangent bundles of EGN$EG_N$

AbstractWe provide a construction of equivariant Lagrangian Floer homology , for a compact Lie group acting on a symplectic manifold in a Hamiltonian fashion, and a pair of ‐Lagrangian submanifolds . We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of . Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent of the auxiliary choices involved in their construction, and are ‐bimodules. In the case when , we show that their chain complex is homotopy equivalent to the equivariant Morse complex of . Furthermore, if zero is a regular value of the moment map and if acts freely on , we construct two ‘Kirwan morphisms’ from to (respectively, from to ). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat ‐connections of a Riemann surface, considered in Manolescu and Woodward's work. Applied to the latter setting, our construction provides an equivariant symplectic side for the Atiyah–Floer conjecture.