Article provides polytopes as dual unit balls of Thurston norms on 3-manifolds.
problem Understanding the dual unit ball shape of Thurston norms.
method Introduced a family of polytopes in Z^2g that can be dual unit balls of Thurston norms on 3-manifolds.
result Polytopes with mod 2 congruent vertices can be realized as dual unit balls of Thurston norms.
Researchers determine the Thurston unit ball for a family of n-chained links and find conditions for fibered faces.
problem Determining the Thurston unit ball and conditions for fibered faces in a family of n-chained links. method Analyzing the family of n-chained links C(n,p), proving the Thurston unit ball is an n-dimensional cocube for p>0, and finding conditions for fibered faces. result The Thurston unit ball for C(n,p) is an n-dimensional cocube for p>0 and provides at least one fibered face for any p. The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.
problem Determining if integral points on the Thurston norm dual ball correspond to geometric structures.
method Examining various geometric, topological, and dynamical structures on 3-manifolds.
result Integral points on the Thurston norm dual ball correspond to the Euler class of taut foliations and other structures.
We simplify Thurston norm computation for 2-bridge link complements.
problem Understanding the complexity of Thurston norm unit balls in 3-manifolds.
method Utilized Floyd and Hatcher's surface description and integral class minimization.
result Thurston norm unit balls of 2-bridge link complements have at most 8 faces.
New flows represent Thurston norm ball faces, differing by veering mutations.
problem Dynamic representation of Thurston norm ball faces by distinct flows.
method Combining veering triangulations and mutations to represent faces by multiple flows.
result Non-fibered faces can be represented by two distinct flows differing by veering mutations.
The Thurston norm of a 3-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of 3-manifolds M with b1(M)=2, and whose fundamental groups admit presentations with two generators and one relator. We show that even among this…
Graph manifolds' Thurston norms are sums of linear functionals, and every such norm can be realized.
problem Understanding Thurston norms of graph manifolds and their realizability.
method Analyzing the structure of Thurston norms as sums of linear functionals and showing realizability.
result Every Thurston norm of a graph manifold can be expressed as a sum of absolute values of linear functionals with rational coefficients.
Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.
problem Characterizing the behavior of volume functions associated with quadratic differentials.
method Analyzing the Thurston volume of unit balls in measured lamination spaces.
result The volume function is not proper and is p-integrable for any 0<p<1. Veering triangulations link Thurston norm and isotopy of surfaces.
problem Understanding the relationship between Thurston norm and isotopy of surfaces.
method Analyzing veering triangulations and their relation to Thurston norm and isotopy.
result Veering triangulations specify faces of Thurston norm balls and link isotopy of surfaces.
The paper extends Thurston's method to new variants of Mather-Thurston theorem.
problem Proving new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
method Generalizing Thurston's technique to prove new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
result The paper answers questions posed by Gelfand-Fuks and Greenberg on PL foliations and Rybicki on contactomorphisms.
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…
Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.
problem Detecting taut polynomials of fibered faces of Thurston norm balls
method Developing a framework for profinite invariance of twisted multivariable Alexander polynomials
result Proving finite quotients detect taut polynomials
Paper calculates ball number of links using Lorentz geometry and circle packing.
problem Calculating the minimum number of balls needed to represent a link.
method Lorentz geometry and circle packing theorem applied to ball packings.
result Shows ball(L)≤5cr(L) for any link L. Algorithm computes Thurston norm for hyperbolic 3-manifolds.
problem Computing the Thurston norm for hyperbolic 3-manifolds.
method Developed a theory of spun-normal immersed surfaces and implemented an algorithm.
result Computed the unit ball of the Thurston norm for cusped hyperbolic 3-manifolds.
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…
New method calculates Thurston norm for 3-manifolds with toroidal boundaries.
problem Computing the Thurston norm for 3-manifolds with toroidal boundaries.
method maw dual graph construction and sutured manifold hierarchies.
result Explicit procedure to compute Thurston norm from hierarchies.
We present a new proof of Thurston's theorem that the unit ball of a seminorm on Rd taking integer values on Zd is a polyhedra defined by finitely many inequalities with integer coefficients.
The paper disproves a conjecture about 3D manifolds using even lattice points.
problem Thurston's Euler class one conjecture for fillable contact structures.
method Analyzing finite covers of hyperbolic 3-manifolds and properties of their dual Thurston norm unit balls.
result Found counter-examples to the conjecture using even lattice points on boundary.
Developed algorithms to compute three polynomial invariants of veering triangulations.
problem Computing polynomial invariants of veering triangulations.
method Introduced and used algorithms for taut, veering, and Teichmüller polynomials based on upper and lower tracks of veering triangulations.
result Proved that the lower and upper taut polynomials are equal but the veering polynomials can differ.
The taut polynomial equals a twisted Alexander polynomial.
problem Understanding the relationship between taut polynomials and Alexander polynomials.
method Defined taut polynomial of veering triangulations and proved it equals a twisted Alexander polynomial.
result The taut polynomial equals a twisted Alexander polynomial of the underlying manifold.
The Whitehead link exterior lacks most Euler class taut foliations.
problem Existence of co-orientable taut foliations with specific Euler classes.
method Combining foliation theory techniques and topological obstructions.
result Most lattice points in the dual Thurston ball cannot be Euler classes of co-orientable taut foliations.
Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.
We highlight several analogies between the Finsler (infinitesimal) properties of Teichmüller's metric and Thurston's asymmetric metric on Teichmüller space. Thurston defined his asymmetric metric in analogy with Teichmüllers' metric, as a solution to an extremal problem, which consists, in the case of the asymmetric me…
Let M be a closed hyperbolic 3-manifold with a fibered face σ of the unit ball of the Thurston norm on H2(M). If M satisfies a certain condition related to Agol's veering triangulations, we construct a taut branched surface in M spanning σ. This partially answers a 1985 question of Oertel, and extends an e…
Investigates flat bundles over low-dimensional manifolds and their cobordism classes.
problem The cobordism of flat bundles over low-dimensional manifolds.
method Study of flat M-bundles over low-dimensional manifolds, comparing a finite dimensional Lie group G with extDiff0(G) and localizing the holonomy. result Flat M-bundles over low-dimensional manifolds are cobordant to a flat M-bundle. About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed manifold, Mostow rigidity assures that a hyperbolic structure is unique when it ex…
Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by non-negative integer weights, 14 for each 3-simplex, that describe how many copies o…
Study of fibred faces of Thurston polyhedra for 2-component 2-bridge links
problem Study of fibred faces of Thurston polyhedra for 2-component 2-bridge links
method Novikov homology associated with the universal covering of the exterior of the link
result Prove that a cohomology class can be represented by a fibration over a circle if and only if its 2-variable Alexander polynomial is ξ-monic We study the space C(a0,a1,…,an) of hyperbolic 2-spheres with cone points of prescribed apex curvatures 2a0,2a1,…,2an∈]0,2π[ and some related spaces. For n=3, we get a detailed description of such spaces. The euclidean 2-spheres were considered by W. P. Thurston: for n=4, the corresponding space…
Study of geometric structures on projective space complement without Schwarz conditions.
problem Geometric structures on projective space complement without Schwarz conditions.
method Use of Dunkl system to study geometric structures.
result Space is a cone-manifold.
Ozsvath and Szabo have defined a knot concordance invariant tau that bounds the 4-ball genus of a knot. Here we discuss shortcuts to its computation. We include examples of Alexander polynomial one knots for which the invariant is nontrivial, including all iterated untwisted positive doubles of knots with nonnegative T…
Solves a triangulation problem by showing minimum tetrahedra equals minimum integral 3-chain.
problem Finding the minimum number of tetrahedra to extend a triangulation of a 2-sphere to a 3-ball.
method Relates the minimum number of tetrahedra to the minimum integral 3-chain norm, proving them equal and showing how to achieve the minimum.
result The minimum number of tetrahedra needed to extend a triangulation of a 2-sphere to a 3-ball equals the minimum integral 3-chain norm.
A polynomial invariant for veering triangulations helps in understanding 3-manifold fibers.
problem Understanding the fibers of 3-manifolds using veering triangulations.
method Introducing a polynomial invariant Vτ associated to veering triangulations and using flow graphs. result The invariant Vτ recovers the Teichmüller polynomial for fibered faces and determines cones in homology. New triangulations encode flows with vanishing polynomial.
problem Constructing veering triangulations with vanishing taut polynomial.
method Using connections between veering triangulations and pseudo-Anosov flows.
result Created arbitrarily large veering triangulations with vanishing taut polynomial.
If a Legendrian knot Λ in the standard contact 3-sphere bounds an orientable exact Lagrangian surface Σ in the standard symplectic 4-ball, then the genus of Σ is equal to the slice genus of (the smooth knot underlying) Λ, the sum of the Thurston-Bennequin number of L and the Euler characteristic of Σ is zero …
After having investigated the regular prisms and prism tilings in the $\SLR$ space in the previous work \cite{Sz13-1} of the second author, we consider the problem of geodesic ball packings related to those tilings and their symmetry groups pq21. $\SLR$ is one of the eight Thurston geometries that can be de…
The paper counts mapping classes by Nielsen-Thurston type, finding growth rates for different subsets.
problem Counting mapping classes in Teichmüller space with different subsets.
method Introduced complexity length to measure negative curvature of curve complexes.
result Growth rates for finite-order, reducible, and multitwists subsets.
Analyzes convex structures in Teichmüller space unit tangent spheres.
problem Characterize faces and extreme points of unit tangent spheres in Teichmüller space.
method Analyzes Finsler infinitesimal balls of Thurston metric, characterizes faces, exposed faces, and extreme points.
result Characterizes faces and extreme points of unit tangent spheres in Teichmüller space.
We consider a hyperbolic surface bundle over the circle with the smallest known volume among hyperbolic manifolds having 3 cusps, so called "the magic manifold". We compute the entropy function on the fiber face of the unit ball with respect to the Thurston norm, determine homology classes whose representatives are gen…
We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston …
This paper illustrates a computational approach to Culler-Morgan-Shalen theory using ideal triangulations, spun-normal surfaces and tropical geometry. Certain affine algebraic sets associated to the Whitehead link complement as well as their logarithmic limit sets are computed. The projective solution space of spun-nor…
Survey on Thurston metric on Teichmüller space, focusing on extremal maps.
problem Constructing extremal Lipschitz maps between hyperbolic surfaces.
method Review of constructions including Thurston's original work.
result Coarse geometry and isometry rigidity of Thurston metric discussed.
A classical result in knot theory says that the Alexander polynomial of a fibered knot is monic and that its degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexa…
Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.
problem Classifying mapping class groups and rational maps.
method Unified proof following Bers' approach.
result Unified proof of Nielsen-Thurston classification.
Geometric interpretation of 3-manifold invariants using immersed curves.
problem Obstructing smooth equivalences between 4-manifolds and surfaces with boundary.
method Relating morphisms between bordered Floer invariants to cobordism maps via immersed curves in the punctured torus.
result Morphisms between immersed curve invariants compute certain cobordism maps.
Developed theory for Thurston maps with a small set of essential singularities.
problem Characterizing Thurston maps with essential singularities.
method Analyzed pullback maps on Teichmüller space to characterize Thurston maps.
result Established a characterization theorem for Thurston maps with four postsingular values.
Article explores Thurston's circle packing theorem in 3-manifold geometry.
problem Understanding Thurston's circle packing theorem in 3-manifold geometry.
method Analyzes the Koebe-Andre'ev-Thurston Theorem and its relation to Thurston's circle packing theorem.
result Illustrates the significance of Thurston's circle packing theorem in 3-manifold geometry.
Overview of Thurston's work in math.
problem None explicitly stated, focuses on Thurston's contributions.
method Presentation of significant results.
result Impact of Thurston's work on mathematics.