arXiv research

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

Paper proves depth bounds for taut foliations using instanton Floer homology.

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…

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

Friedl and Kim show any taut sutured manifold can be realized as a twisted homology product, but their proof gives no practical description of how complicated the realizing representation needs to be. We give a number of results illustrating the relationship between the topology of a taut sutured handlebody and the com…

For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology ($SFH$) can be used to determine all fibered classes in $H^1(M)$. Furthermore, we show that the $SFH$ of a balanced sutured manifold $(M,γ)$ detects which classes in $H^1(M)$ admit a taut depth one foliatio…

For closed 3-manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3-manifold Y, there is a bijection between vertices of the HF^+(Y) polytope carrying the group Z and the faces of the Thurston norm unit ball that corresp…

Using the virtual fibering theorem of Agol we show that a sutured 3-manifold $(M, R_+,R_-,γ)$ is taut if and only if the $\ell^2$-Betti numbers of the pair $(M,R_-)$ are zero. As an application we can characterize Thurston norm minimizing surfaces in a 3-manifold $N$ with empty or toroidal boundary by the vanishing of …

A 3-manifold is foliar if it supports a codimension-one co-oriented taut foliation. Suppose $M$ is an oriented 3-manifold with connected boundary a torus, and suppose $M$ contains a properly embedded, compact, oriented, surface $R$ with a single boundary component that is Thurston norm minimizing in $H_2(M, \partial M)…

We survey the interactions between foliations and contact structures in dimension three, with an emphasis on sutured manifolds and invariants of sutured contact manifolds. This paper contains two original results: the fact that a closed orientable irreducible 3-manifold M with nonzero second homol-ogy carries a hyperti…

We prove that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber …

Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if (M, γ)--> (M', γ') is a sutured manifold decomposition then SFH(M',γ') is a direct…

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…

The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic oriented cusped $3$-manifolds can be split in a "symmetrization" factor and a "reduced" state sum. We show that these factors are invariants on their own, that we call "symmetry defects" and "reduced QHI", provided the manifolds are endowed w…

New method calculates Thurston norm for 3-manifolds with toroidal boundaries.

New method finds unique branched surfaces in 3-manifolds.

In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decompos…

We study when the Thurston norm is detected by twisted Alexander polynomials associated to representations of the 3-manifold group to SL(2, C). Specifically, we show that the hyperbolic torsion polynomial determines the genus for a large class of hyperbolic knots in the 3-sphere which includes all special arborescent k…

Framework extends sutured manifold invariants to pairs with torus boundaries.

Developed a real sutured Heegaard Floer theory.

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…

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…

Generates sutured manifolds from a surface using surgery.

Abstract TQFT for sutured manifolds using Floer homology.

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…

Study on tautness tensor for Riemannian foliations.

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.

New framework for sutured manifolds using handleslides and Heegaard invariants.

Sutured instanton homology uses Heegaard diagrams to bound homology dimensions.

3-manifolds with Heegaard 2 admit taut foliations if their fundamental group is left-orderable.

David Gabai showed that disk decomposable knot and link complements carry taut foliations of depth one. In an arbitrary sutured 3-manifold M, such foliations F, if they exist at all, are determined up to isotopy by an associated ray [F] issuing from the origin in H^1(M;R) and meeting points of the integer lattice H^1(M…

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…

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…

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

Taut foliations map leaves to branched 2-sphere covers.

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…

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

New homological action on sutured instanton homology defined.

We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. A linking curve is defined as a curve in the tangent space whose composition with the exponential map is tree formed. This key idea is used to define sutured manifolds. We prove first that any sutured manifold sati…

Left orderability proven for certain 3-manifolds with specific foliations.

We give an explicit construction of the Honda--Kazez--Matić gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link F…

Tautness of submanifolds in spheres is preserved under Lie sphere transformations.

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…

In this paper, we generalize the work of the second author and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handle-bo…

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.

A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions. The aim of this paper is to demonstrate that taut ideal triangulations are very common, and that their behavio…

This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite di…

Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.