New patterns deform Farey triangulation in symmetric space.
problem Deforming Farey triangulation in symmetric space.
method Realized representations of modular group as isometry groups of geodesic patterns in SL3(R)/SO(3). result 2-parameter family of deformations of Farey triangulation.
Study quasisymmetric maps on hyperbolic plane boundaries.
problem Identify quasisymmetric maps corresponding to specific lambda lengths and flip distances.
method Analyze maps on Farey triangulation, relate to shearing coordinates and flip distance.
result Identify quasisymmetric maps corresponding to pinched lambda lengths and flip distances.
Study of q-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.
problem Geometry of q-rationals and their properties. method Construction and analysis of deformed Farey triangulation and deformed modular surface; definition and study of Springborn operations.
result Derivation of a formula for the q-deformed midpoint and new q-deformation of Markov numbers. Paper develops geometry for Kleinian groups using Farey polynomials.
problem Understanding the geometry of Kleinian groups generated by parabolic elements.
method Sakuma-Weeks triangulations and Farey recursive polynomials.
result Simple recursive algorithm to determine link complement geometry.
Classifies positive integral friezes on surfaces.
problem Classifying positive integral friezes on marked bordered surfaces.
method One-to-one correspondence with ideal triangulations and rescaling constants.
result Number of non-equivalent friezes on bordered surfaces is finite.
Geodesic patterns, shears, and Anosov representations of the modular group.
problem Understanding representations of the modular group into Isom(X).
method Analyzing geodesic patterns, shears, and foliations.
result The Barbot component is homeomorphic to R^2 x [0,∞), with interior and boundary properties.
Paper defines Farey Recursive Functions and explores their properties.
problem Understanding recursive functions on rationals.
method Defined and studied Farey Recursive Functions using Farey graph.
result Farey Recursive Functions naturally connect to 2-bridge knots and links.
Study finds polynomial convergence rate for Farey sequences linked to Riemann hypothesis.
problem Understanding convergence rates of maximum mean discrepancies for Farey sequences.
method Identifying positive-semidefinite kernels and their polynomial convergence rates.
result Polynomial convergence rate of maximum mean discrepancies of Farey sequences is equivalent to the Riemann hypothesis.
Every infinitely edge-connected graph has a minor of Farey graph or Tℵ0∗t.
problem Characterizing edge-connected graphs with specific minor properties.
method Analyzing the minor structure of infinitely edge-connected graphs.
result Infinitely edge-connected graphs contain Farey graph or Tℵ0∗t as a minor. The paper generalizes Farey tessellation to 3D hyperbolic space.
problem Generalizing Farey tessellation to higher dimensions.
method Introducing conformal bryophylla and classifying them.
result Properties of conformal bryophylla's limiting sets studied.
Study calculates stable norm of slit tori using Farey sequence.
problem Computing the stable norm of slit tori.
method Explicit computations using the Farey sequence and gluing slit tori.
result Estimates the asymptotic counting of simple homology classes.
The Farey tree helps embed rational balls and lens spaces into complex projective space.
problem Embedding rational homology balls and lens spaces into complex projective space.
method Recursive Kirby calculus argument using the Farey tree.
result Explicit constructions of embeddings of triples of rational homology balls into homotopy CP2. Optimal Farey sequence for Γ0(2n) with upper bound 2n−1.
problem Finding an optimal Farey sequence for the congruence subgroup Γ0(2n). method Proving the existence of a Farey sequence with specific properties and uniqueness.
result The upper bound of the Farey sequence is optimal and equals 2n−1. We describe the (P)SL(2,C) character varieties of all 2-bridge knots and the diagonal character varieties for all 2-bridge links in terms of a set of polynomials defined using Farey recursion.
Octagon map accelerates diagonal changes algorithm.
problem Improving the efficiency of diagonal changes algorithm.
method Octagon Farey map as an acceleration.
result Octagon map accelerates diagonal changes algorithm.
A new proof shows how to characterize maps using simple geometry.
problem Characterizing quasisymmetric maps on the unit circle.
method Elementary proof using normal family argument and hyperbolic geometry.
result Characterizes quasisymmetric maps via shear coordinates on the Farey tesselation.
The study of Farey polynomials connects geometry, topology, and combinatorics.
problem Understanding the combinatorics of Farey polynomials and their applications.
method Recursive definition of Farey polynomials, combinatorial analysis, and geometric/topological connections.
result New properties and recursive definition of Farey polynomials, providing practical solutions to classification problems.
Study Agol cycles in pseudo-Anosov 3-braids.
problem Understanding conjugacy invariants of pseudo-Anosov maps.
method Investigate train tracks associated with Farey intervals and describe Agol cycles.
result Complete description of Agol cycles in pseudo-Anosov 3-braids.
Study geodesics in 3-torus, determining complements' topology.
problem Understanding the topology of geodesic complements in 3-torus.
method Analyzes the orbit of direction vectors under PSL3(Z) action and uses Farey graph distances. result Determines homeomorphism type of geodesic complements in 3-torus.
We show that for a surface S, the subgraph of the pants graph determined by fixing a collection of curves that cut S into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made by Aramayona, Parlier, and Shackleton and has…
We give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation.
Connected graph for twice-punctured torus curves.
problem Structure of tri-pants graph on twice-punctured torus.
method Examined relationship with Farey complex to prove connectivity and infinite diameter.
result Tri-pants graph is connected and has infinite diameter.
With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze the k-Farey graphs Fk and F⩽k, two natural variants of the Farey graph in which we relax the edge condition to indicate intersection number =k or ≤k, respectively. The former, $\…
Our main theorem asserts that every Farey graph embedded in the 1-skeleton of the pants complex of any finite type surface is totally geodesic.
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…
There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…
Classifies 3-braids from choreographic motions on Lissajous curves, linking them to mapping classes and geodesics.
problem Classifying 3-braids from choreographic motions on Lissajous curves.
method Parametrization in terms of levels and slopes, using dilatation and geodesic cutting sequences.
result Dilatation of pseudo-Anosov mapping classes increases with level or slope.
Study of circle homeomorphisms with square summable diamond shears.
problem Characterizing circle homeomorphisms with specific summability properties.
method Analysis of homeomorphisms in modular coordinates and comparison to Weil-Petersson class.
result Sharp results comparing new class to Weil-Petersson class and Hölder classes.
Study jigsaw constructions of hyperbolic lattices and answer questions on arithmeticity and pseudomodularity.
problem Constructing and analyzing non-commensurable, non-uniform, non-arithmetic lattices in hyperbolic geometry.
method Hyperbolic jigsaw construction and recursive formulas for tessellations.
result Demonstration of recursive formula for tessellation of hyperbolic plane, generalizing Farey addition.
Our main theorem identifies a class of totally geodesic subgraphs of the 1-skeleton of the pants complex, each isomorphic to the product of two Farey graphs. We deduce the existence of many convex planes in the 1-skeleton of the pants complex.
Study of SU(2,1) character varieties on one-holed torus.
problem Characterize representations of mapping class group on SU(2,1) character variety.
method Explicit description of SU(2,1) character variety, use of Farey graph adaptation, and mapping class group action analysis.
result Description of an open domain of discontinuity for mapping class group action.
We parametrize the space Z of Zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the Farey tesselation. Then we express the Hilbert transform and the Fourier coefficients of the Zygmund vector fields in terms of the above parametrization by infinitesimal shear functions. F…
The paper studies algebraic integer relations and sequences converging to 4.
problem Investigating algebraic integer relations and convergence of sequences.
method Constructing a generalized Farey graph for the subgroup Gα and analyzing its properties. result A sequence of algebraic integers converges to 4, each corresponding to a non-free group of rank 2.
We use the geometry of the Farey graph to give an alternative proof of the fact that if A∈GL2Z and GA=Z2⋊AZ is generated by two elements, there is a single Nielsen equivalence class of 2-element generating sets for GA unless A is conjugate to $\pm \left(\begin {smallma…
Connected flip graphs for triangulations on hyperbolic surfaces.
problem Connecting triangulations on hyperbolic surfaces via flips.
method Proving connectedness of flip graphs and giving bounds on edge flips.
result Flip graphs of geometric triangulations are connected.
Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight t…
New isolated geometric triangulations found in once-punctured torus bundles.
problem Identifying isolated geometric triangulations in 3-manifolds.
method Examining ideal triangulations and their moves to find isolated geometric ones.
result Infinite family of once-punctured torus bundles with isolated geometric triangulations.
Efficient triangulations help in understanding 3-manifold boundaries.
problem Understanding boundary slopes in 3-manifolds.
method Introducing and studying boundary-efficient triangulations and inflating ideal triangulations.
result There are only finitely many boundary slopes for incompressible and \(\partial\)-incompressible surfaces in compact 3-manifolds.
A 6-regular triangulation for hyperbolic plane created.
problem Creating a 6-regular triangulation for hyperbolic plane.
method Constructed a 6-regular geodesic triangulation.
result A 6-regular geodesic triangulation of the hyperbolic plane was successfully created.
The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
Geometrically, twist numbers on punctured tori are dense and non-continuous.
problem Understanding twist numbers on hyperbolic punctured tori.
method Hyperbolic geometry and Farey graph analysis.
result The graph of twist numbers is dense in [0,1]x[0,1].
A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens…
Agol recently introduced the concept of a veering taut triangulation, which is a taut triangulation with some extra combinatorial structure. We define the weaker notion of a "veering triangulation" and use it to show that all veering triangulations admit strict angle structures. We also answer a question of Agol, givin…
Minimal triangulations for 229 hyperbolic census knots discovered.
problem Finding minimal triangulations for hyperbolic census knots.
method Ideal triangulations of the magic manifold, low-complexity triangulations for partial fillings, sorting into families.
result Minimal triangulations for 229 hyperbolic census knots discovered, conjectured to be minimal for all 42 families.
We construct a Poincaré section for the horocycle flow on the modular surface SL(2,R)/SL(2,Z), and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…
A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In…
Authors find small triangulations for specific 4-manifolds.
problem Finding optimal triangulations for 4-manifolds.
method Triangulated connected sums of CP^2 and S^2×S^2, conjectured minimal pentachora.
result Triangulations have the smallest number of pentachora for their types.
Combinatorial description of 3-manifolds using ordered triangulations.
problem Understanding closed 3-manifolds through ideal triangulations.
method Combining ordered ideal triangulations and Pachner moves.
result Closed 3-manifolds can be described via ordered triangulations and moves.