arXiv research

New invariants improve Heegaard Floer slice genus and clasp number bounds.

The paper uses Heegaard Floer homology to study homology cobordism groups.

We make use of the action of $H_1(Y)$ in Heegaard Floer homology to generalize the Ozsváth-Szabó correction terms for $3$-manifolds with standard $\operatorname{HF}^\infty$. We establish the basic properties of these invariants: conjugation invariance, behavior under orientation reversal, additivity, and spin$^c$ ratio…

The Heegaard Floer correction term ($d$-invariant) is an invariant of rational homology 3-spheres equipped with a Spin$^c$ structure. In particular, the correction term of 1-surgeries along knots in $S^3$ is a ($2\mathbb{Z}$-valued) knot concordance invariant $d_1$. In this paper, we estimate $d_1$ for the $(2,q)$-cabl…

We derive a closed formula for the Heegaard Floer correction terms of lens spaces in terms of the classical Dedekind sum and its generalization, the Dedekind-Rademacher sum. Our proof relies on a reciprocity formula for the correction terms established by Ozsvath and Szabo. A consequence of our result is that the Casso…

We use Heegaard Floer homology with twisted coefficients to define numerical invariants for arbitrary closed 3-manifolds equipped torsion spin$^c$ structures, generalising the correction terms (or $d$--invariants) defined by Ozsváth and Szabó for integer homology 3-spheres and, more generally, for 3-manifolds with stan…

We discuss a concordance invariant constructed from Heegaard Floer homology "correction terms" and +/- 1 surgeries on knots in the three-sphere.

Study on $Δ$ invariants of plumbed 3-manifolds, comparing with Heegaard-Floer homology.

Study shows infinite rank in bipolar filtration of topologically slice knots.

Study homology cobordism and plumbed manifolds using Heegaard Floer homology.

In this remark, we show how the monopole Frøyshov invariant, as well as the analogues of the Involutive Heegaard Floer correction terms $\underline{d},\overline{d}$, are related to the $\mathrm{Pin}(2)$-equivariant Floer homology $\mathit{SWFH}^G_*$. We show that the only interesting correction terms of a $\mathrm{Pin}…

Study of knot Floer homology and its relation to Heegaard Floer homology via equivariant surgery.

We show that for rational surface singularities with odd determinant the mu-bar invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4-ball. We identify the mu-bar invariant with the corresponding correction term in Heegaard Floer theory.

For a fixed p, there are only finitely many elliptic 3-manifolds given by p/q-surgery on a knot in S^3. We prove this result by using the Heegaard Floer correction terms (d-invariants) to obstruct elliptic manifolds from arising as knot surgery.

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

Turaev showed that there is a well-defined map assigning to an oriented link L in the three-sphere a Spin structure t_0 on Sigma(L), the 2-fold cover of S^3 branched along L. We prove, generalizing results of Manolescu-Owens and Donald-Owens, that for an oriented quasi-alternating link L the signature of L equals minus…

Heegaard Floer homology duality applied to 4D mapping tori.

Classifies 3-manifolds bounding rational homology balls.

Given an element in the first homology of a rational homology 3-sphere $Y$, one can consider the minimal rational genus of all knots in this homology class. This defines a function $Θ$ on $H_1(Y;\mathbb Z)$, which was introduced by Turaev as an analogue of Thurston norm. We will give a lower bound for this function usi…

Holomorphic polygons help calculate link complements' Floer homology.

We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involu…

Study Heegaard Floer homology for two-component L-space links with zero linking number.

We compute the Pin(2)-equivariant monopole Floer homology for the class of plumbed 3-manifolds with at most one "bad" vertex (in the sense of Ozsvath and Szabo). We show that for these manifolds, the Pin(2)-equivariant monopole Floer homology can be calculated in terms of the Heegaard Floer/monopole Floer lattice compl…

Heegaard Floer homology bounds Thurston norm for certain knots and links.

Floer homology linked to Milnor fibers for certain singularities.

The paper studies Heegaard Floer homology for manifolds with torus boundaries and proves properties.

Ozsvath and Szabo defined an analog of the Froyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover of the 3-sphere branched over a knot K, we obtain an invariant delta of knot concordance. We show that delta is determined by the signature for alt…

We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold Sigma x [0,1] x R, where Sigma is the Heegaard surface, instead of Sym^g(Sigma). We then show that the entire invariance proof can be carried out in our setting. In the process, we derive a new formula for the index of the…

Cabling explained using curves on a punctured torus.

Study Heegaard Floer homology and word metric on Torelli group.

We derive new obstructions to periodicity of classical knots by employing the Heegaard Floer correction terms of the finite cyclic branched covers of the knots. Applying our results to two fold covers, we demonstrate through numerous examples that our obstructions are successful where many existing periodicity obstruct…

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

We use the Heegaard-Floer homology correction terms defined by Ozsváth--Szabó to formulate a new obstruction for a knot to be of finite order in the smooth concordance group. This obstruction bears a formal resemblance to that of Casson and Gordon but is sensitive to the difference between the smooth versus topological…

Real Heegaard Floer theory shown to be natural and invariant.

New formulas link Milnor invariants to Heegaard Floer homology.

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

Study shows 3D shapes can't fit in certain 4D spaces.

Let $K$ be a null-homologous knot in a three-manifold $Y$. We give a description of the Heegaard Floer homology of integer surgeries on $Y$ along $K$ in terms of the filtered homotopy type of the knot invariant for $K$. As an illustration, we calculate the Heegaard Floer homology groups of non-trivial circle bundles ov…

We show that the correction terms in Heegaard Floer homology give a lower bound to the the genus of one-sided Heegaard splittings and the $\mathbb Z_2$--Thurston norm. Using a result of Jaco--Rubinstein--Tillmann, this gives a lower bound to the complexity of certain closed $3$--manifolds. As an application, we compute…

We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with $\underline{d}(Y) \neq d(Y) \neq \overline{d}(Y)$. We also construct a homomorphism from the three-dimensional homolo…

New invariants from Heegaard Floer homology for knot concordance.

We compute the Heegaard Floer homology of $S^3_1(K)$ (the (+1) surgery on the torus knot $T_{p,q}$) in terms of the semigroup generated by $p$ and $q$, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath--Szabo d-invariant. We relate the result to known knot invariants of $T_{p,q}$ as …

We show that there are links whose individual components are concordant to the unknot, but which are not concordant to any link with unknotted components. We give examples in the topological category, and examples in the smooth category which are topologically slice. We also give generalizations regarding components of…

We consider a stabilized version of hat Heegaard Floer homology of a 3-manifold Y (i.e. the U=0 variant of Heegaard Floer homology for closed 3-manifolds). We give a combinatorial algorithm for constructing this invariant, starting from a Heegaard decomposition for Y, and give a combinatorial proof of its invariance pr…

Real Heegaard Floer Homology extends Li's real monopole Floer homology.

Paper proves a conjecture about a Heegaard Floer invariant for certain rational homology spheres.

We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descr…

Study proves obstructions to equivariantly slice strongly negative amphichiral knots.