Finite presentation for a specific group in 3D handlebody topology.
problem Finding a finite presentation for a specific group in 3D handlebody topology.
method Constructed a finite presentation for the liftable Hilden group to derive the presentation for the balanced superelliptic handlebody group.
result A finite presentation was given for the balanced superelliptic handlebody group.
Study of liftable mapping class group for superelliptic covers.
problem Understanding mapping class groups of superelliptic covers.
method Computational and algebraic methods to study the liftable mapping class group.
result The liftable mapping class group is independent of the degree of the cover and has finite abelianization.
Finite presentations for mapping class groups of surfaces and surfaces with points/boundaries.
problem Finding finite presentations for balanced superelliptic mapping class groups.
method Construct finite presentations for corresponding liftable mapping class groups in a different generating set.
result Finite presentations for balanced superelliptic mapping class groups of various surfaces.
Three elements generate balanced superelliptic mapping class groups.
problem Generating balanced superelliptic mapping class groups.
method Proving groups are generated by three elements through normalizers and liftable mapping class groups.
result Balanced superelliptic mapping class groups are generated by three elements.
Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.
problem Characterize Veech groups of infinite superelliptic curves.
method Analyzing geometric properties, differential equations, and group theory.
result Veech groups of infinite superelliptic curves are all matrices permuting branched points.
Proves a minimal generating set for a specific group of mapping classes.
problem Finding a minimal generating set for a specific group of mapping classes.
method Proved the group is generated by four elements, with minimal exceptions.
result Minimal generating set for the balanced superelliptic mapping class group.
The reduced Burau representation Vn of the braid group Bn is obtained from the action of Bn on the homology of an infinite cyclic cover of the disc with n punctures. The group homology H∗(Bn;Vn) of braid groups with coefficients in the complexified reduced Burau representation is calculated. Our topolog…
Study of superelliptic curves and complex braid groups, computing homology.
problem Understanding the homology of superelliptic curves and their associated braid groups.
method Analysis of the universal family of superelliptic curves, computation of homology groups using complex braid groups.
result Complete calculation of integral homology groups of End over finite fields. Positive factorization found for a specific map on surfaces.
problem Balanced superelliptic rotation on surfaces.
method Positive factorization approach.
result Positive factorization for balanced superelliptic rotation.
Market maker handles negative prices with unique asset swapping.
problem Handling negative prices in financial markets.
method Unique market mechanism with numeraire currency, liquidity extensions.
result Liquidity fingerprint and payoff compared to established models.
The paper studies liftable mapping class groups of cyclic covers of spheres.
problem Understanding liftable mapping class groups of cyclic covers of spheres.
method Derived finite generating sets, provided algorithms, determined isomorphism classes, derived presentations, and calculated normalizers and centralizers.
result Presentations and isomorphism classes of liftable mapping class groups for various covers.
The Loch Ness Monster admits many regular dessins d'enfants and different holomorphic structures.
problem Classical theory of dessins d'enfants on compact surfaces extended to non-compact surfaces.
method Study of infinite genus surfaces and their connections to Riemann surfaces.
result The Loch Ness monster admits infinitely many regular dessins d'enfants.
The paper constructs braiding structures for a specific subfactor.
problem The challenge is to understand the braiding structures of a Jones-Wassermann subfactor.
method The approach involves constructing braiding structures on the multi-interval Jones-Wassermann subfactor planar algebra.
result The braiding structures induce a projective unitary representation of the balanced superelliptic mapping class group.
In this paper we propose and investigate a novel nonlinear unit, called Lp unit, for deep neural networks. The proposed Lp unit receives signals from several projections of a subset of units in the layer below and computes a normalized Lp norm. We notice two interesting interpretations of the Lp unit. First…
Study covers of surfaces, showing types and properties.
problem Characterize homeomorphism types of surface covers.
method Analyzing homeomorphism types of covers of surfaces, identifying universal covers and homology covers.
result Identify four possible homeomorphism types for certain covers of surfaces.
Study noncommutative coverings of irrational quantum tori.
problem Characterize noncommutative coverings of irrational quantum tori.
method Developed a framework for noncommutative coverings and studied irrational quantum tori.
result Characterized all connected noncommutative coverings of irrational quantum tori.
A {\em solvable} cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of l…
After showing that a covering space of surface bundles over S1 factors as a `covering of fibers' followed by a `power covering', we prove that, for torus bundles, power coverings do not lower Heegaard genus, and that fiber coverings lower the genus only in special cases.
Covering preserves bottom spectrum, implies amenable covering.
problem Spectral preservation in Riemannian coverings.
method Proving spectral properties of Schrödinger operators on coverings.
result Covering preserving bottom spectrum implies amenability.
Characterizes sphere covers with lifted homeomorphisms.
problem Characterizing sphere covers with lifted homeomorphisms.
method Characterization through cyclic branched covers and homeomorphisms.
result Every homeomorphism of the sphere lifts to a homeomorphism of the covering surface in these cases.
In "Rips complexes and covers in the uniform category" \cite{Rips} the authors define, following James \cite{J}, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given…
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.
Criteria found for knots not determined by their double branched covers.
problem Determining knots from their double branched covers.
method Criteria based on tunnel number one knots and their double branched covers.
result Provided criteria for knots not determined by their double branched covers.
A foliation of a manifold M is called R-covered if its lift to the universal cover of M has space of leaves R. We show that there are many graph manifolds which admit taut foliations, but which do not admit any R-covered foliations. On the other hand, we show that these manifolds all have finite covers admitting R-cove…
Develops a unified theory for various covering structures in noncommutative geometry.
problem Creating a general theory for different types of coverings in noncommutative geometry.
method Formulates a set of axioms applicable to all types of coverings in the list.
result Provides algebraic proofs for topological and differential geometric results.
The paper studies which branched covers can be lifted to braided embeddings.
problem Which branched covers lift to braided embeddings?
method Analyzes conditions for liftability using Hansen's criterion and examples in different dimensions.
result Not all branched covers lift to braided embeddings; examples are provided in various dimensions.
We discuss construction of coverings of the unit ball of a finite dimensional Banach space. The well known technique of comparing volumes gives upper and lower bounds on covering numbers. This technique does not provide a construction of good coverings. Here we apply incoherent dictionaries for construction of good cov…
The paper details folding of branched covers of the 3-sphere over knots.
problem Understanding folding of branched covers of the 3-sphere over knots.
method Presenting detailed foldings of dihedral covers of the 3-sphere branched over torus knots.
result Detailed foldings of dihedral covers of the 3-sphere branched over torus knots are presented.
We study tori which are cyclic covers of the standard torus, that is, the deck transformation group of the covering map is cyclic. These covering tori can be parametrized in a natural way and we show that being cyclic is equivalent to certain arithmetic condition on these parameters. There is a natural $\mathrm{SL}(2,\…
Course on knots using branched coverings.
problem Determining knots using cyclic branched coverings.
method Cyclic branched coverings of knots.
result New insights into knot classification.
Second part of a series on higher coverings of racks and quandles.
problem Characterizing higher-dimensional centrality conditions in racks and quandles.
method Applying higher categorical Galois theory to racks and quandles.
result Identification and characterization of higher coverings, trivial coverings, and normal coverings.
Study Milnor invariants for covering links to distinguish certain links.
problem Distinguish links using Milnor invariants for covering links.
method Generalize Hartley and Murasugi's covering linkage invariants and use cobordism invariants.
result First non-vanishing Milnor invariants of a Brunnian link modulo 2 equals a sum of linking numbers of covering links.
Study fundamental groups of small covers and their injective submanifolds.
problem Topology of small covers and their fundamental groups.
method Explicit presentations of fundamental groups and combinatorial data analysis.
result Characterization of 3D small covers with nonnegative scalar curvature.
In this paper we consider completed coverings that are branched coverings in the sense of Fox. For completed coverings between PL manifolds we give a characterization of the existence of a monodromy representation and the existence of a locally compact monodromy representation. These results stem from a characterizatio…
The paper studies self-covers of manifolds and their algebraic structures.
problem Understanding the algebraic structure of manifolds with self-covers.
method Analyzes self-covers of manifolds and their induced linear endomorphisms.
result Establishes an algebraic structure theorem for the fundamental group of manifolds.
Gromov-Thurston covers have Betti numbers as expected.
problem Understanding Betti numbers of branched covers of hyperbolic manifolds.
method Analyzing Gromov-Thurston branched covers and their Betti numbers.
result Betti numbers match expectations for non-divisible degree covers.
Formula compares metrics on branched coverings of line bundles.
problem Comparing metrics on branched coverings of line bundles.
method Formula to compare Quillen metrics on branched coverings.
result Formula for comparing Quillen metrics on branched coverings.
Covering spectra match if the covering is amenable, with conditions on curvature.
problem Matching spectra of Riemannian coverings under amenability conditions.
method Analyzing spectra of Riemannian manifolds and their coverings under completeness and curvature constraints.
result Spectra match if the covering is amenable, with conditions on curvature.
Study how bottom of spectra changes with Riemannian coverings.
problem Behavior of bottom of spectra under Riemannian coverings.
method Analysis of scalar Schrödinger operators on Riemannian manifolds.
result Changes in the bottom of spectra observed under coverings.
Study of lifting maps in branched covers of 3-manifolds, showing non-injectivity.
problem Injectivity of lifting maps in branched covers of 3-manifolds.
method Analysis of branched covers and mapping class groups.
result Lifting map is generally not injective for most regular branched covers of 3-manifolds.
Super tau-covers extend bihamiltonian hierarchies' symmetries.
problem Extending symmetries of bihamiltonian hierarchies.
method Constructing super tau-covers for bihamiltonian integrable hierarchies.
result Symmetries of bihamiltonian hierarchies extended to super tau-covers.
Paper defines embolic volume and relates it to Betti number using the covering trick.
problem Relating embolic volume to topological invariants.
method Covering trick from systolic geometry applied to Berger's inequality.
result Relates embolic volume to the first Betti number.
Characterizes Riemannian orbifolds and their coverings via metric geometry.
problem Understanding Riemannian orbifolds and their coverings.
method Characterizes Riemannian orbifolds and their coverings via metric geometry.
result The metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and the natural projection is an orbifold covering.
Given a finite cover f:tilde{G} \to G and an embedding of tilde{G} in the plane, Negami conjectures that G embeds in P^2. Negami proved this conjecture for regular covers. In this paper we define two properties (Propserties V and E), depending on the cover tilde{G} and its embedding into S^2, and generalize Negami's re…
Study the spectrum of covering solenoids from manifold sequences.
problem Understanding the spectrum of covering solenoids from manifold sequences.
method Examined the leaf-wise Laplacian on covering solenoids and related it to the spectra of individual manifolds in the sequence.
result The spectrum of the Laplacian on the covering solenoid equals the closure of the union of the spectra of the manifolds in the sequence.
Study of collapsed manifolds with bounded Ricci curvature and non-collapsed universal cover.
problem Understanding collapsed manifolds with specific Ricci curvature properties.
method Ricci flow techniques applied to non-collapsed universal cover.
result Partial extension of nilpotent structural results to global Ricci bounded covering geometry.
Study on spectral stability of Riemannian coverings.
problem Stability of eigenvalues in Riemannian coverings.
method Analysis of Laplacian eigenvalues under finite coverings.
result Necessary conditions for spectral stability or instability.
Study involutions on 3D small covers, proving quotient spaces are linked 2-spheres.
problem Understanding quotient spaces of 3D small covers by involutions.
method Analyzing quotient spaces and using topological properties.
result Quotient spaces of 3D small covers by involutions are linked 2-spheres.