Study centers of mapping-torus groups to define knot and mapping class invariants.
problem Understanding the center of mapping-torus groups.
method Determine the center of meta-nilpotent quotients of mapping-torus groups.
result Introduce two invariants of knots and mapping classes as quadratic forms.
Quantum trace map defined for 3-manifolds with torus boundaries.
problem Quantifying topological structures of 3-manifolds with torus boundaries.
method Defining a quantum trace map from skein module to a quantum torus module.
result Established a 3D quantum trace map for 3-manifolds with torus boundaries.
We develop a method to find a set of diminimal polyhedral maps on the torus from which all other polyhedral maps on the torus may be generated by face splitting and vertex splitting. We employ this method, though not to its completion, to find 53 diminimal polyhedral maps on the Torus.
Study nearly parallel G2-structures with torus symmetry using multi-moment maps.
problem Characterize and construct nearly parallel G2-structures with torus symmetry.
method Use multi-moment map techniques and analyze the geometry of the base spaces.
result Locally, the construction may produce examples with four-torus symmetry.
Hamiltonian cycles found in toroidal maps.
problem Hamiltonicity of doubly semi-equivelar maps on the torus.
method Analyzing 2-uniform tilings of the plane to derive Hamiltonian cycles.
result Every doubly semi-equivelar map on the torus contains a Hamiltonian cycle.
Biharmonic maps are generalizations of harmonic maps. A well-known result of Eells and Wood on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree ±1. It would be interesting to know if there …
If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map X on the torus is a quotient of an Archimedean tiling on the plane then the map…
Semi-Equivelar maps are generalizations of maps on the surfaces of Archimedean solids to surfaces other than the 2-sphere. The well known 11 types of normal tilings of the plane suggest the possible types of semi-equivelar maps on the torus and the Klein bottle. In this article we classify (up to isomorphism) semi-eq…
Embeddings of mapping tori for end-periodic graph maps are proven.
problem Embedding mapping tori of end-periodic graph maps into finite complexes.
method Flowline-preserving homotopy equivalence and π1-injective map. result Every mapping class of Γ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.
Each closed oriented 3-manifold M is naturally associated with a set of integers D(M), the degrees of all self-maps on M. D(M) is determined for each torus bundle and torus semi-bundle M. The structure of torus semi-bundle is studied in detail. The paper is a part of a project to determine D(M) for all 3-ma…
The mapping class group of a Heegaard splitting is the group of connected components in the set of automorphisms of the ambient manifold that map the Heegaard surface onto itself. For the genus three Heegaard splitting of the 3-torus, we find an eight element generating set for this group. Six of these generators induc…
We study the effect of the mapping class group of a reducible 3-manifold M on each incompressible surface that is invariant under a self-homeomorphism of M. As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures: A reducible 3-manifold admits an Anosov torus if an…
Researchers compute zeta-determinants and analytic torsion for metric mapping tori.
problem Computing zeta-determinants and analytic torsion for metric mapping tori.
method Using the BFK-gluing formula for zeta-determinants.
result Computed zeta-determinants and analytic torsion for metric mapping tori.
Semi-Equivelar maps are generalizations of Archimedean solids to the surfaces other than 2-sphere. There are eight semi-equivelar maps of types {33,42}, {32,4,3,4}, {6,3,6,3}, {34,6}, {4,82}, {3,122}, {4,6,12}, {6,4,3,4} exist on the torus. In this article we show the e…
Study proves rigidity of harmonic maps from 2-torus to complex projective space.
problem Rigidity of isotropic harmonic maps from a 2-torus to a complex projective space.
method Proves rigidity through holomorphic embeddings and complete linear systems.
result Ensures rigidity of harmonic bands in condensed matter physics.
New flow connects symplectic maps to hyperKähler geometry.
problem Understanding symplectic maps and their geometry.
method Established a correspondence between symplectic diffeomorphisms and hyperKähler moment maps.
result Introduced a new flow, the modified moment map flow.
Researchers calculate dimensions of skein modules for 2-torus mapping tori.
problem Determining dimensions of Kauffman bracket skein modules for specific cases.
method Using generic q and decomposing twisted Hochschild homology of G-skein algebras. result Dimensions of skein modules for G=SL2 and G=GL1 are calculated. Proves geodesic connections on 2-torus without invariant tori.
problem Existence of geodesic connections on 2-torus without invariant tori.
method Uses J. Mather's result on connecting orbits for monotone twist maps.
result Proves existence of connecting geodesics on unit tangent bundle of 2-torus.
Study Agol cycles on 2-punctured torus and 5-punctured sphere, finding new dilatation formula.
problem Understanding Agol cycles on specific surfaces.
method Computed measured train tracks and Agol cycles for pseudo-Anosov maps.
result Found a new formula for the dilatation of pseudo-Anosov maps.
Minimal maps from surfaces to torus found for various genus values.
problem Finding minimal degree maps from genus g surfaces to the torus. method Constructing simplicial degree d maps from a triangulation of a genus g surface to the 7-vertex triangulation of the torus. result Minimal maps exist for g≥1 and ∣d∣≥2g−1 for g≥3. The study explores maps of 2- and 3-uniform tilings on the torus.
problem Understanding the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
method Analyzing the quotient maps of 2- and 3-uniform tilings on the torus.
result Bounds on the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
In this note, we show that, if a pseudo-Anosov map φ:S→S admits a finite cover whose action on the first homology has spectral radius greater than 1, then the monodromy of any fibered structure of any finite cover of the mapping torus Mφ has the same property.
Authors prove quantum invariant conjecture for figure-eight knot complement.
problem Connecting quantum invariants of surface diffeomorphisms to hyperbolic volumes.
method Analyzes the simplest case of a one-puncture torus and figure-eight knot complement.
result Proves conjecture linking quantum invariant to hyperbolic volume.
We show that the cone associated with a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers provided the dimension of the torus is bigger than 2 and that no orbit is tangent to the contact distribution. This may be consider…
This study proves the local existence of a symplectic gradient flow on a flat torus.
problem Proving the local existence of a symplectic gradient flow on a flat torus.
method Using a moment map and a DeTurck trick to make the flow strictly parabolic and showing local existence and regularity.
result The group of symplectomorphisms of the real four-dimensional torus is locally contractible.
Classifies 2-uniform maps on torus with formulas and asymptotic bounds.
problem Classifying 2-uniform maps on torus.
method Classification through arithmetic functions and asymptotic analysis.
result Explicit formulas and asymptotic bounds for 2-uniform maps on torus.
We prove that for every P there is a bound B depending only on P so that the mapping torus of every P--small irreducible train-track map can be obtained by surgery from one of B mapping tori. We show that given an integer P>0 there is a bound M depending only on P, so that there exists a presentation of the fundament…
Quotients of torus endomorphisms have parabolic orbifolds.
problem Understanding the structure of quotients of torus endomorphisms.
method Analyzing the properties of torus endomorphisms and their quotients.
result Every quotient of a torus endomorphism has a parabolic orbifold.
The study determines Z2-Thurston norms in Sol manifolds and embeds non-orientable surfaces.
problem Determining Z2-Thurston norms in Sol manifolds and embedding non-orientable surfaces. method Analyzing the action of torus maps on curve complexes and constructing incompressible surfaces.
result Determination of Z2-Thurston norms and embeddability of non-orientable surfaces in Sol manifolds. New theorem on flat tori stability using harmonic maps and Ricci flow.
problem Stability of flat tori under Ricci and scalar curvature bounds.
method Harmonic map heat flow, Ricci flow, and RCD theories.
result Gromov-Hausdorff stability theorem for flat 3-tori.
New findings show the Gilmer-Masbaum map isn't always one-to-one.
problem Determining the injectivity of the Gilmer-Masbaum map on Kauffman bracket skein modules.
method Computed the image of the evaluation map for specific cases of mapping tori and analyzed homology classes.
result The restriction of the Gilmer-Masbaum map to certain homology classes is not injective.
The study improves bounds on pseudo-Anosov maps and certifies minimum and accumulation points of normalized dilatations.
problem Understanding the set of normalized dilatations of fully-punctured pseudo-Anosov maps.
method Improving bounds on the number of tetrahedra in veering triangulations and using computational means.
result Certified that the minimum element of the set of normalized dilatations is μ2 and the minimum accumulation point is μ4. Smooth torus actions on moduli spaces of super stable curves and maps of genus zero.
problem Constructing smooth C∗-actions on moduli spaces of super stable curves and maps of genus zero. method Using the implicit function theorem, proving smooth split atlases, and studying automorphism groups.
result Explicit descriptions of normal bundles to fixed loci in terms of spinor bundles and sections.
We give two generalizations of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action: 1) replacing the torus action by a compact connected Lie group action, 2) replacing the manifold having a torus action by an equivariant map. This provides a systematic method for calcula…
The paper develops a method to map knots in a cylinder to virtual-flat knots.
problem How to map knots in a cylinder to virtual knots.
method Construct a diagram on a cylinder with invisible crossings, then pull back invariants.
result Developed a method to map knots in a cylinder to virtual-flat knots.
We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants…
Partial proof of a conjecture about knot concordance maps.
problem Proving a conjecture about homomorphisms in knot concordance.
method Analyzing self-maps of the knot concordance group.
result Proved a map is not a homomorphism for certain winding numbers.
A vertex-transitive map X is a map on a closed surface on which the automorphism group Aut(X) acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse…
Quantum theory uses modular group representations to assign invariants to 3-manifolds.
problem Assigning invariants to 3-manifolds via modular group representations.
method Projective representations of the modular group derived from a noncommutative torus.
result Computed traces and determinants of matrices associated with modular group elements.
Study on (λ,λ)-eigenfunctions on compact manifolds, showing manifold properties and eigenfamily dimensions.
problem Characterizing compact manifolds with (λ,λ)-eigenfunctions and understanding their eigenfamilies. method Analyzing (λ,λ)-eigenfamilies on compact Riemannian manifolds, showing that any such manifold is a mapping torus and any (λ,λ)-eigenfamily is one-dimensional. result Any compact manifold admitting a (λ,λ)-eigenfunction is a mapping torus and any (λ,λ)-eigenfamily is one-dimensional. Lower bound on volumes of special mapping tori.
problem Calculating the minimum volume of compactified mapping tori.
method Using strongly irreducible end-periodic homeomorphisms and properties of pants graphs.
result Volume of compactified mapping tori is comparable to the translation length of the homeomorphism on pants graphs.
Given a reducible 3-manifold M with an aspherical summand in its prime decomposition and a homeomorphism f:M→M, we construct a map of degree one from a finite cover of M⋊fS1 to a mapping torus of a certain aspherical 3-manifold. We deduce that M⋊fS1 has virtually infinite first Be…
We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types {36}, {44}, {63}, {33,42}, {32,4,3,4}, {3,6,3,6}, {34,6}, {4,82}, $\…
The paper identifies Borsuk-Ulam property for specific map classes between torus and Klein bottle.
problem Determining Borsuk-Ulam property for map classes between torus and Klein bottle.
method Using homotopy classes and free involutions of spaces, the paper analyzes fundamental groups to find the property.
result Homotopy classes of maps from torus to Klein bottle with specific involutions have the Borsuk-Ulam property.
New invariants explain topological properties of pseudo-Anosov maps.
problem Understanding invariants of pseudo-Anosov maps at roots of unity.
method Numerical methods and quantum modularity conjecture.
result Descendant invariants related by Fourier transform.
Study shows Heegaard genus relation in 3-manifold amalgamation.
problem Understanding Heegaard genus in 3-manifold amalgamation.
method Examined amalgamation of 3-manifolds, used degree-one maps, and fixed boundary conditions.
result Proved g(M)≥g(N), showing Heegaard genus relation. Semi-Equivelar maps are generalizations of Archimedean solids to the surfaces other than 2-sphere. In earlier work a complete classification of semi-equivelar map of type (35,4) on the surface of Euler characteristic -1 was given. In the meantime Karabas an Nedela classified vertex transitive semi-equivelar maps on…
We prove that the mapping torus group $\FN \rtimes_α \Z$ of any automorphism α of a free group $\FN$ of finite rank n≥2 is weakly hyperbolic relative to the canonical (up to conjugation) family H(α) of subgroups of $\FN$ which consists of (and contains representatives of all) conjugacy classes that …