arXiv research

Introduces integer-valued Heegaard Floer theory with canonical orientations.

Heegaard Floer theory is a kind of topological quantum field theory, assigning graded groups to closed, connected, oriented 3-manifolds and group homomorphisms to smooth, oriented 4-dimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with ext…

Developed a real sutured Heegaard Floer theory.

New signs and gradings enable detailed comparison in Heegaard Floer theory.

Proves a formula in Heegaard Floer homology using combinatorial methods.

Extends Heegaard Floer theory to surfaces of dimension one.

Lecture notes on Heegaard Floer homology for beginners.

We review the use of grid diagrams in the development of Heegaard Floer theory. We describe the construction of the combinatorial link Floer complex, and the resulting algorithm for unknot detection. We also explain how grid diagrams can be used to show that the Heegaard Floer invariants of 3-manifolds and 4-manifolds …

Study proves naturality and functoriality in a type of Heegaard Floer homology.

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

Study shows link polynomial evaluations from Heegaard Floer theory.

Survey on proof of homology isomorphism between two complex theories.

Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As a part of the proof, we also establish combinatorially the invariance of the linear-categorical representation of the strongly-based mapping class groupoid given by the same theor…

New findings on 3-manifolds using Heegaard Floer theory.

In a previous paper, Sarkar and the third author gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected three-manifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boun…

Combinatorial method computes Legendrian knot invariant.

This is an expansion on my talk at the Geometry and Topology conference at McMaster University, May 2004. We outline a program to relate the Heegaard Floer homologies of Ozsvath-Szabo, and Seiberg-Witten-Floer homologies as defined by Kronheimer-Mrowka. The center-piece of this program is the construction of an interme…

This paper provides an introduction to the basics of Heegaard Floer homology with some emphasis on the hat theory and to the contact geometric invariants in the theory. The exposition is designed to be comprehensible to people without any prior knowledge of the subject.

Floer homology linked to Milnor fibers for certain singularities.

Proves properties of instanton knot Floer homology and connected sum formula.

Real Heegaard Floer homology gets a new grading for certain 3-manifolds.

A three-manifold equipped with a Heegaard diagram can be used to set up a Floer homology theory whose differential counts pseudo-holomorphic disks in the $g$-fold symmetric product of the Heegaard surface. This leads to a topological invariant for three-manifolds, Heegaard Floer homology, which is functorial under cobo…

This paper uses Heegaard Floer theory to study pseudo-Anosov flows and their periodic points.

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

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

This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston…

We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule define…

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

Characterizes $(1,1)$ knots with large Dehn surgeries.

Defines real link Floer homology for specific types of links.

In this paper we show how to recover the relative Q-grading in Heegaard Floer homology from the noncommutative grading on bordered Floer homology.

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

Researchers link knot Floer homology, Burau representation, and quantum gl(1|1).

Ozsvath and Szabo have established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link L in S^3 and the Heegaard Floer homology of its double-branched cover. This relationship has since been recast by the authors as a specific instance of a broa…

It follows implicitly from recent work in Heegaard Floer theory that lens spaces are homology cobordant exactly when they are oriented homeomorphic. We provide a new combinatorial proof using the Heegaard Floer d-invariants, which themselves may be defined combinatorially for lens spaces.

We define a new homology theory we call symbol homology by using decorated moduli spaces of Whitney polygons. By decorating different types of moduli spaces we obtain different flavors of this homology theory together with morphisms between them. Each of these flavors encodes the properties of a different type of Heega…

New surgery exact triangles in Heegaard Floer homology for rational slopes.

Extends Khovanov homology spectral sequence using Heegaard Floer homology.

Heegaard Floer homology study confirms composition maps match up to homotopy.

Defines new Heegaard Floer invariants with actions of both E and F.

New technique connects instanton Floer homology to Heegaard diagrams for knots and 3-manifolds.

An excision theorem connects Heegaard Floer homology of 3-manifolds.

Let A be a dg algebra over F_2 and let M be a dg A-bimodule. We show that under certain technical hypotheses on A, a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product of M with itself and converges to the Hochschild homology of M. We apply th…

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.

Identifies Heegaard Floer homology solid tori via Dehn fillings.

New framework for sutured manifolds using handleslides and Heegaard invariants.

We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete answer to the geography problem for components of peculiar modules of tangles. As a main application, we show that Conway mut…

Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we develop a model for $\mathbb{Z}/2$-equivariant symplectic Floer theory using equivariant almost complex structures, which admits a localization map to a twisted version of Floer cohomology in the invariant set. We then present applications to S…