arXiv research

New findings on 3-manifolds using Heegaard Floer theory.

Research classifies L-space knots using Floer simple manifolds.

Graph manifold L-space intervals computed and applied to cables.

The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.

A criterion for Whitney disks connects intersections in 3-manifold homology.

We show that if a positive integral surgery on a knot K inside a homology sphere X with Seifert genus g(K) results in an induced knot K_n in X_n(K)=Y which has simple Floer homology, we should have n>=2g(K). Moreover, if X is the standard sphere, the three-manifold Y is a L-space and the Heegaard Floer homology groups …

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…

If a 3--manifold $Y$ contains a non-separating sphere, then some twisted Heegaard Floer homology of $Y$ is zero. This simple fact allows us to prove several results about Dehn surgery on knots in such manifolds. Similar results have been proved for knots in $L$--spaces.

New proof shows Mazur-type manifolds with specific boundaries are simple.

An oriented three-manifold with torus boundary admits either no L-space Dehn filling, a unique L-space filling, or an interval of L-space fillings. In the latter case, which we call "Floer simple," we construct an invariant which computes the interval of L-space filling slopes from the Turaev torsion and a given slope …

Enhances Floer homology for fibrations without simple connectivity assumption.

We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. More…

We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the holomorphic disk Floer homologies for three-manifolds defined in an earlier paper. We set up basic properties of these invariants, including an Euler characteristic calculation, behaviour under connected sums. Then, we …

Study slopes on knot manifolds to understand their fundamental groups.

Study of Pin(2)-monopole Floer homology under connected sums.

Study Brieskorn spheres using Floer homology, generating infinite rank summands in homology cobordism.

The paper classifies a special family of knots in lens spaces using knot Floer homology.

We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It carries information about the Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive inf…

Paper introduces a new invariant for knots in contact 3-manifolds.

Motivated by conjectures in Heegaard Floer homology, we introduce an invariant HC(Y) of the cohomology ring of a closed 3-manifold Y whose behavior mimics that of the Heegaard Floer homology HF^\infty(Y,s) for s a torsion spin-c structure. We derive from this a numerical invariant h(Y), and obtain upper and lower bound…

The paper examines 2-torsion in instanton Floer homology for knots and 3-manifolds.

We determine the Fukaya Floer homology of the three-manifold which is the product of a Riemann surface of genus $g\geq 1$ times the circle. This sets up the groundwork for finding the structure of the Donaldson invariants of four-manifolds not of simple type in the future. We give the following applications: 1) We show…

Any Haken 3--manifold (possibly with boundary consisting of tori) can be transformed into a $\mathrm{surface}\times S^1$ by a series of splitting and regluing along incompressible surfaces. This fact was proved by Gabai as an application of his sutured manifold theory. The first half of this paper provides a few techni…

Positive braid knots have simple knot Floer homology.

We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge's construction of knots in the three-sphere which admit lens space surgeries is complete. The first step, which we prove here, is to show that a knot in a lens space with a three-sphere surgery has simple (in…

New proof of Heegaard Floer surgery formulas using Fukaya category of the torus.

Develops Floer theory for 3-manifold covers using equivariant structures.

We show that if K is a non-trivial knot inside a homology sphere X, the rank of the knot Floer homology group associated with K is strictly bigger than the rank of the Heegaard Floer homology group associated with X.

Paper classifies Heegaard Floer minimal knots in sutured manifolds.

Floer theories provide invariants for 3- and 4-manifolds.

Formula for Heegaard Floer multicurves of double tangles from knot complements.

Lecture notes on Heegaard Floer homology for 3-manifolds and knots.

Combinatorial definition of bordered Floer theory for torus-boundary manifolds.

Geometrically generates Fukaya categories of Weinstein manifolds.

Study shows infinitely many non-simple Legendrian knots in two-bridge knots.

New knots are found to be non-simple in Legendrian contact geometry.

Survey of bordered Floer homology for 3-manifolds with boundary.

Injective maps in various link homologies induced by ribbon concordances proved.

Developed a real sutured Heegaard Floer theory.

For a rational homology 3-sphere $Y$ with a $\spinc$ structure $\s$, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology lead to a collection of variants which are topological invariants. We establish exact sequences relating them, we show that they satisfy a dua…

Study shows inequality in Floer homologies for 3-manifold covers.

Lower bounds on rational slice genus using Heegaard Floer invariants.

For a word w in the braid group on n-strands, we denote by T_w the corresponding transverse braid in the rotational symmetric tight contact structure on S^3. We exhibit a map on link Floer homology which sends the transverse invariant associated to T_{ws_i} to that associated to T_w, where s_i is one of the standard ge…

Link Floer homology operations and basepoint moves are analyzed.

Introduces Floer lasagna modules using link Floer homology.

This is the fourth of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an a…

Develops Floer cohomology for 4-manifolds with involutions and links.

Real Heegaard Floer theory shown to be natural and invariant.