Constructs taut branched surfaces from veering triangulations in hyperbolic 3-manifolds.
problem Constructing taut branched surfaces in hyperbolic 3-manifolds.
method Using veering triangulations and Agol's conditions.
result Constructs a taut branched surface spanning a fibered face in a hyperbolic 3-manifold.
Taut foliations map leaves to branched 2-sphere covers.
problem Understanding taut foliations in 3-manifolds.
method Map foliation leaves to branched 2-sphere covers.
result Taut foliations are characterized by such maps.
A formula calculates the Euler class of foliations using dual graphs.
problem Calculating the Euler class of foliations using cooriented branched surfaces.
method Using dual graphs of cooriented branched surfaces to define a simplicial 1-cycle representing the Poincaré dual of the Euler class.
result The formula generalizes previous results and classifies realizable homology classes.
We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental group, and are not L-spaces.
Left orderability proven for certain 3-manifolds with specific foliations.
problem Proving left orderability for specific 3-manifolds with taut foliations.
method Analyzing the fundamental group of a 3-manifold with a co-orientable taut foliation.
result The fundamental group of the 3-manifold is left orderable.
The paper explores left-orderable groups in branched cyclic covers with taut foliations.
problem Left-orderability of fundamental groups in branched cyclic covers.
method Study of left-orderability in cyclic branched covers of links with co-oriented taut foliations.
result Proves left-orderability for specific types of branched covers and Dehn surgeries.
Defines new lamination links in 3-manifolds and shows their properties.
problem Understanding lamination links in 3-manifolds.
method Introduces and defines oriented framed measured lamination links, showing their relationship to Seifert laminations.
result Taut Seifert laminations generalize minimal genus Seifert surfaces.
New method finds unique branched surfaces in 3-manifolds.
problem Finding unique branched surfaces in 3-manifolds.
method Proving endperiodic maps have periodic splitting sequences.
result Veering branched surfaces are unique up to equivalence.
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…
We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology, and can therefore be used to study the question of which foliations arise as geometric limits …
Let F be a leafwise hyperbolic taut foliation of a closed 3-manifold M and let L be the leaf space of the pullback of F to the universal cover of M. We show that if F has branching, then the natural action of π1(M) on L is faithful. We also show that if F has a finite branch locus B whose stabilize…
Classifies Anosov flows on figure-eight knot surgeries.
problem Classifying Anosov flows on Dehn surgeries on the figure-eight knot.
method Combining Plante's classification of M(0) with Schwider's branched surfaces.
result M(r) carries a unique Anosov flow if r is an integer, and no Anosov flow if r is not an integer.
Suppose that F is a transversely oriented, codimension one foliation of a connected, closed, oriented 3-manifold. Suppose also that F has continuous tangent plane field and is {\sl taut}; that is, closed smooth transversals to F pass through every point of M. We show that if $\mathcal…
New foliations constructed from contact pairs, revealing flexible taut foliations.
problem Characterizing taut foliations in 3D.
method Construction of codimension-one foliations from pairs of contact structures in 3D.
result Foliations constructed are taut, providing new insights into the L-space conjecture.
The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.
problem Analyzing the geometric and combinatorial effects of smoothing intersections in arcs or curves.
method Geometric and combinatorial analysis, proving tautness and arc length spectrum properties.
result Shortest arcs with self-intersections have exactly or at most one more self-intersection than the self-intersection number.
Detect slopes in toroidal 3-manifolds to prove properties of fundamental groups.
problem Prove properties of fundamental groups of toroidal 3-manifolds.
method Slope detection using left-orders, foliations, and Heegaard Floer homology.
result Toroidal integer homology spheres have left-orderable fundamental groups.
Simplified proof of a theorem about 3D shapes.
problem Proving a theorem about foliations on 3-manifolds.
method Using foliated branched covers.
result A simple proof of Novikov's theorem.
Paper tackles L-space conjecture for knot manifolds, proving equivalence for some properties.
problem Tackles L-space conjecture for knot manifolds, proving equivalence for some properties. method Introduces relative L-space conjecture, characterizes slope detection, uses Heegaard Floer homology, left-orders, and foliations. result Confirms equivalence of CTF and NLS for slope detected knots, identifies exceptional slopes. 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…
Persistently foliar knots have a unique foliation property under certain surgeries.
problem Characterizing knots with a specific foliation property under surgeries.
method Analyzing foliations in knot complements and using Dehn surgery.
result Composite knots and nontrivial connected sums of fibered knots are persistently foliar.
3-manifolds foliar if surfaces minimize Thurston norm, leading to taut foliations.
problem Understanding foliar 3-manifolds via minimizing surfaces.
method Defining double-diamond taut sutured manifolds and analyzing their implications.
result Many slopes are strongly realized in foliations of 3-manifolds, including knot complements.
Let M be a fibered 3-manifold with multiple boundary components. We show that the fiber structure of M transforms to closely related transversely oriented taut foliations realizing all rational multislopes in some open neighborhood of the multislope of the fiber. Each such foliation extends to a taut foliation in t…
We show that $|MS(L_1 # L_2)|=|MS(L_1)|\times|MS(L_2)|\times\mathbb{R}$ when L1 and L2 are any non-split and non-fibred links. Here MS(L) denotes the Kakimizu complex of a link L, which records the taut Seifert surfaces for L. We also show that the analogous result holds if we study incompressible Seifert s…
The paper proves left-orderability for certain Dehn fillings of pseudo-Anosov mapping tori.
problem Left-orderability of fundamental groups in Dehn fillings of pseudo-Anosov mapping tori.
method Two approaches: one using R-covered foliations and the other using one-sided branching. result All such Dehn fillings have left-orderable fundamental groups.
Sutured manifolds have zero L2-Betti numbers if and only if they are taut.
problem Characterizing taut sutured 3-manifolds.
method Using the virtual fibering theorem and ℓ2-Betti numbers. result Taut sutured 3-manifolds have zero ℓ2-Betti numbers. The article proves properties of Seifert links and their cyclic branched covers.
problem Properties of Seifert links and their cyclic branched covers.
method Left-orderability, co-oriented taut foliations, and L-space properties. result Proves the ADE link conjecture for Seifert links. We continue the study of linear families of contact forms on 3-manifolds begun in our paper `Contact geometry and complex surfaces'. The present paper introduces Teichmuller and moduli spaces for so-called taut contact circles. By constructing a developing map for taut contact circles, we show that these geometrically …
This paper proves a converse to Thurston's 1976 observation for taut foliations on hyperbolic 3-manifolds.
problem Proving a converse to Thurston's 1976 observation for taut foliations on hyperbolic 3-manifolds.
method Analyzing the Euler class and using hyperbolic geometry properties.
result For a taut foliation on a hyperbolic 3-manifold, if the Euler characteristic of a closed leaf equals the Euler class, then there exists another taut foliation with the same Euler class.
In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a neg…
We define a laminar branched surface to be a branched surface satisfying the following conditions: (1) Its horizontal boundary is incompressible; (2) there is no monogon; (3) there is no Reeb component; (4) there is no sink disk (after eliminating trivial bubbles in the branched surface). The first three conditions are…
For surface-knots, branched covers with degree 3 have simplifying numbers <3.
problem Understanding numerical invariants of branched covering surface-knots.
method Analyzing numerical invariant (simplifying number) of branched covering surface-knots with degree 3.
result Branched covering surface-knots with degree 3 have simplifying numbers less than 3.
We present a few general results on foliations of 4-manifolds by surfaces: existence, tautness, relations to minimal genus of embedded surfaces; as well as some open problems. We hope to stimulate interest in this area.
Simplifies surface-knots using chart moves involving black vertices.
problem Simplifying branched covering surface-knots.
method Chart moves involving black vertices to simplify surface-knots.
result Properties of simplified branched covering surface-knots with branch points.
Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of represe…
Proofs for decomposing branched affine surfaces into triangles and cylinders.
problem Decomposing branched affine surfaces into simpler geometric shapes.
method Proof of Veech's theorem and introduction of invariant α.
result Any pair of decompositions can be connected by flips.
Study on tautness tensor for Riemannian foliations.
problem Understanding tautness properties of Riemannian foliations.
method Investigating a symmetric 2-tensor related to mean curvature.
result Prove a tautness condition for compact manifolds.
Veering branched surfaces help construct geodesic flows on curved surfaces.
problem Constructing geodesic flows on negatively curved surfaces.
method Introduce veering branched surfaces and surgeries, then use them to construct veering triangulations that correspond to geodesic flows.
result Explicit constructions of veering branched surfaces corresponding to geodesic flows on negatively curved surfaces.
We give an equivalent description of taut submanifolds of complete Riemannian manifolds as exactly those submanifolds whose normal exponential map has the property that every preimage of a point is a union of submanifolds. It turns out that every taut submanifold is also Z2-taut. We explicitely construct gen…
Simplifies surface-knots by adding 1-handles with chart loops.
problem Complexity of branched covering surface-knots.
method Adding 1-handles with chart loops to simplify charts.
result Properties of simplified charts are investigated.
Hyperbolic surface bundles over the circle have minimal entropy close to 1/g.
problem Understanding the minimal entropy of hyperbolic surface bundles over the circle.
method Proved the minimal entropy behavior of hyperbolic surface bundles as branched double covers of the 3-sphere.
result Minimal entropy of hyperbolic surface bundles over the circle behaves like 1/g.
Researchers compute specific Hurwitz numbers for branched covers.
problem Computing the number of equivalence classes of surface branched covers.
method Combinatorial method based on Gronthendieck's dessins d'enfant.
result Explicit arithmetic formulae for weak Hurwitz numbers are derived.
Proves PU(n,1) is 1-taut, concluding studies of rank-one Lie groups.
problem Tautness of rank-one Lie groups of non-compact type.
method Proves PU(n,1) is 1-taut. result Concludes the study of 1-tautness of rank-one Lie groups of non-compact type. We define sink marks for branched complexes and find conditions for them to determine a branched surface structure. These will be used to construct branched surfaces in knot and tangle complements. We will extend Delman's theorem and prove that a Montesinos knot K of length at least 3 has a persistently laminar branc…
Quantized Coulomb branches linked to skein algebras.
problem Understanding the relationship between quantized Coulomb branches and skein algebras.
method Association of quantized Coulomb branches to surfaces, description of relationship for specific surfaces, formulation of a conjecture.
result A conjecture linking quantized Coulomb branches and skein algebras.
The paper studies branched surfaces and their properties.
problem Global topologies of branched surfaces and their maps.
method Explicit construction of maps and study of topological properties.
result New insights into the geography of branched surfaces and their maps.
The number of Klein-bottle leaves in taut foliations is invariant under smooth deformations.
problem Invariance of Klein-bottle leaf counts in taut foliations.
method Proving invariance of the parity of Klein-bottle leaf counts under smooth deformations.
result The parity of Klein-bottle leaf counts is invariant under smooth deformations.
Floer homology detects right-veering monodromy in fibered knots.
problem Detecting right-veering monodromy in fibered knots.
method Using Heegaard Floer homology and symplectic Floer homology.
result Knot Floer homology characterizes tight contact structures.
Tautness of submanifolds in spheres is preserved under Lie sphere transformations.
problem Invariance of tautness under Lie sphere transformations.
method Extending tautness definition to Legendre submanifolds and using Lie sphere transformations to show invariance.
result Tautness is invariant under Lie sphere transformations.