arXiv research

New results show how complex representations are needed to realize taut sutured handlebodies as homology products.

Constructs foliations from Heegaard diagrams for any genus.

New proof of Laudenbach and Poénaru's theorem on 4D 1-handlebodies.

We demonstrate how to combinatorially calculate the EH-class of a compatible contact structure in the sutured Floer homology group of a balanced sutured three manifold which is associated to an abstract partial open book decomposition. As an application we show that every contact three manifold (closed or with convex b…

The paper detects fiberedness in 3-manifolds and extends a torsion invariant to sutured manifolds.

Brunnian theta curves in 3D spheres have hyperbolic exteriors.

A Seifert surface F for a knot K is disk decomposable if there is a taut sutured manifold heirarchy for the complement of F, whose decomposing surfaces are all disks. It follows that F has minimal genus for the knot K, and has handlebody complement, i.e., F is free. We show that these necessary conditions for disk deco…

Framework extends sutured manifold invariants to pairs with torus boundaries.

We define a "sutured topological quantum field theory", motivated by the study of sutured Floer homology of product 3-manifolds, and contact elements. We study a rich algebraic structure of suture elements in sutured TQFT, showing that it corresponds to contact elements in sutured Floer homology. We use this approach t…

Developed a real sutured Heegaard Floer theory.

Sutured instanton homology uses Heegaard diagrams to bound homology dimensions.

Study tight contact structures on Seifert surfaces, finding their number and properties.

Abstract TQFT for sutured manifolds using Floer homology.

Study contact structures and sutured monopole Floer homology, computing and deriving formulas.

Paper constructs maps for sutured monopole Floer homology.

New homological action on sutured instanton homology defined.

We define a sutured cobordism category of surfaces with boundary and 3-manifolds with corners. In this category a sutured 3-manifold is regarded as a morphism from the empty surface to itself. In the process we define a new class of geometric objects, called bordered sutured manifolds, that generalize both sutured 3-ma…

Researchers construct gluing maps and prove duality in sutured Floer homology.

The idea of a sutured topological quantum field theory was introduced by Honda, Kazez and Matić (2008). A sutured TQFT associates a group to each sutured surface and an element of this group to each dividing set on this surface. The notion was originally introduced to talk about contact invariants in Sutured Floer Homo…

Kronheimer and Mrowka defined invariants of balanced sutured manifolds using monopole and instanton Floer homology. Their invariants assign isomorphism classes of modules to balanced sutured manifolds. In this paper, we introduce refinements of these invariants which assign much richer algebraic objects called projecti…

Paper generalizes sutured Floer homologies with new algorithms and polytopes.

We give a necessary and sufficient criterion for a sutured manifold to be taut in terms of the twisted homology of the sutured manifold.

New framework for sutured manifolds using handleslides and Heegaard invariants.

Generates sutured manifolds from a surface using surgery.

Paper shows Khovanov homology isn't preserved by certain mutations.

We study the sutured Floer homology invariants of the sutured manifold obtained by cutting a knot complement along a Seifert surface, R. We show that these invariants are finer than the "top term" of the knot Floer homology, which they contain. In particular, we use sutured Floer homology to distinguish two non-isotopi…

We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of one sutured manifold (M',Γ') into a larger sutured manifold (M,Γ), together with a contact structure on M-M'. As an application of this gluing map, we produce a (1+1)-dimensional TQFT by dimensional reduction and study its…

Contact gluing maps are shown to be equivalent in sutured Floer homology.

We show that the sutured Khovanov homology of a balanced tangle in the product sutured manifold D x I has rank 1 if and only if the tangle is isotopic to a braid.

New strategy proves Ambrose conjecture for 3-manifolds using linking curves and sutured manifolds.

This paper studies knots in sutured manifolds achieving minimum instanton homology rank.

We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with $\Z$ coefficients, of certain sutured manifolds of the form $(Σ\times S^1, F \times S^1)$ where $Σ$ is an annulus or punctured torus. Using this classification, we give a new proof that the contact invariant in sutured Fl…

We show that sutured embedded contact homology is a natural invariant of sutured contact 3-manifolds which can potentially detect some of the topology of the space of contact structures on a 3-manifold with boundary. The appendix, by C. H. Taubes, proves a compactness result for the completion of a sutured contact 3-ma…

For sutured 3-manifolds M, there is a sutured Thurston norm due to Scharlemann. We show how depth one foliations of M and corresponding fibrations and the usual Thurston norm on the double of M are useful tools for computing this norm. In many examples, the faces of the unit ball of the sutured norm are related to cone…

We show that if a surgery on a knot in a product sutured manifold yields the same product sutured manifold, then this knot is a 0-- or 1--crossing knot. The proof uses techniques from sutured manifold theory.

Attaching a 2-handle to a genus two or greater boundary component of a 3-manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surfa…

Proves equivalence of contact invariants in two Floer homology theories.

Study shows spectral sequence linking two periodic braids' homologies.

Study handles in sutured manifolds and knots, finding varied handle numbers and unique surfaces.

We define a torsion invariant T for every balanced sutured manifold (M,g), and show that it agrees with the Euler characteristic of sutured Floer homology SFH. The invariant T is easily computed using Fox calculus. With the help of T, we prove that if (M,g) is complementary to a Seifert surface of an alternating knot, …

Sutured manifolds have zero $L^2$-Betti numbers if and only if they are taut.

New homology theory for annular knots and links.

We exhibit the first example of a knot in the three-sphere with a pair of minimal genus Seifert surfaces that can be distinguished using the sutured Floer homology of their complementary manifolds together with the Spin^c-grading. This answers a question of Juhász. More precisely, we show that the Euler characteristic …

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 F…

We develop monopole and instanton Floer homology groups for balanced sutured manifolds. Applications include a new proof of Property P for knots.

This article is a standalone introduction to sutured Floer homology for graduate students in geometry and topology. It is divided into three parts. The first part is an introductory level exposition of Lagrangian Floer homology. The second part is a construction of Heegaard Floer homology as a special, and slightly mod…

In this paper we construct a Floer-homology invariant for a natural and wide class of sutured manifolds that we call balanced. This generalizes the Heegaard Floer hat theory of closed three-manifolds and links. Our invariant is unchanged under product decompositions and is zero for nontaut sutured manifolds. As an appl…

Quantum invariant constructed for sutured 3-manifolds using Hopf superalgebra.