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.
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.
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.
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.
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.
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…
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. 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. 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. 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.
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. 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.
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.
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.
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.
Identifies half-space neighborhoods of pleating rays in the Riley slice of Schottky groups.
problem Determining points in the Riley slice of Schottky groups.
method Adapting ideas from L. Keen and C. Series, identifying half-space neighborhoods of pleating rays.
result Provides a provable method to determine if a point is in the Riley slice.
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.
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.
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…
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].
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…
We present a complete classification of elements in the mapping class group of the torus which have a representative that can be written as a product of two orientation reversing involutions. Our interest in such decompositions is motivated by features of the monodromy maps of real fibrations. We employ the property th…
A musical instrument based on moduli spaces lets users hear geometric concepts.
problem Understanding geometric structures through auditory means.
method Developed a plastic hormonica based on Farey tessellations and Poincaré disk decorations.
result Users can audibly experience paths in Riemann moduli spaces and listen to mapping classes.
Proves effective slope gaps for lattice surfaces.
problem Proving effective slope gaps for lattice surfaces.
method Proves effective slope gap distribution for square torus and general lattice surfaces.
result Effective slope gap distribution result for lattice surfaces.
The paper proves a free product decomposition for congruence subgroups with constraints.
problem Proving a free product decomposition for congruence subgroups with specific constraints.
method Using the convex hull of the extended Farey sequence and properties of the hyperbolic plane.
result Established bounds and characterizations for minimum denominators in cusp sets.
New tree structure for pseudo-Anosovs from interval maps.
problem Understanding pseudo-Anosovs from interval maps.
method Tree structure on pseudo-Anosovs using rational numbers.
result Deepened dictionary between invariants.
Spheres in curve graphs are connected, proving Gromov boundary linearity.
problem Understanding connectivity in curve graphs and their boundaries.
method Defining spheres and analyzing their connectivity for different complexities.
result Spheres in high complexity curve graphs are always connected, with weaker results for low complexity.
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.
We give a parametrization to the asymptotic Teichmuller space of the open unit disk through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Then using the parametrization, we define a new metric on the asymptotic Teichmuller space. Two other rela…
This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is…
Study tunnel numbers of cable knots and their companions, proving new bounds and constructing examples.
problem Understanding the relationship between the tunnel numbers of a knot and its cable.
method Combinatorial techniques and analysis of Heegaard splittings.
result Proves that for many cases, the tunnel number of a cable knot equals the original knot's tunnel number plus one.
For every half-translation surface with marked points (M,Σ), we construct an associated tessellation Π(M,Σ) of the Poincaré upper half plane whose tiles have finitely many sides and area at most π. The tessellation Π(M,Σ) is equivariant with respect to the action of PSL(2,R), and invariant w…
New F-polynomial distinguishes knotoid diagrams not previously possible.
problem Polynomial invariants for knotoids.
method Introduced F-polynomial and constructed distinguishing examples. result New invariant F distinguishes knotoids not previously possible.