In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.
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48 results for “Dihedral covers”
Formula for dihedral cover signatures of four-manifolds.
problem Signature of dihedral covers of four-manifolds.
method Formula derivation and necessary conditions for intersection forms.
result Sharp necessary condition for two-bridge slice singularities.
Study dihedral covers of four-manifolds with singularities, proving only a slice knot can arise.
problem Understanding dihedral covers of four-manifolds with singularities.
method Lifting trisection of a surface to a branched cover, computing signature contributions.
result Only a slice knot can arise as a unique singularity on irregular dihedral covers of .
Dihedral Linking Invariantsmath.GT
Dihedral linking invariant uses knot colorings to distinguish knots.
problem Distinguishing knots using knot colorings and linking numbers.
method Algorithm for computing linking numbers in dihedral branched covers.
result The dihedral linking invariant distinguishes more than 98% of prime knot pairs.
Algorithm calculates ribbon obstructions for colored knots.
problem Computing ribbon obstructions for colored knots.
method Algorithm based on dihedral branched covers and colored knot diagrams.
result Algorithm successfully computes ribbon obstructions for colored knots.
The Dihedral Genus of a Knotmath.GT
The dihedral genus of a knot is related to its signature and minimal surface genus.
problem Determining the dihedral genus of knots and its relation to signatures and minimal surface genus.
method Using dihedral branched covers and properties of Fox colorings, the dihedral genus is linked to the signature of the cover and the Murasugi signature of the knot.
result An infinite family of knots with minimal genus surfaces that realize the four-genus of the knot.
Algorithm calculates linking numbers in three-manifolds.
problem Computing linking numbers in three-manifolds.
method Algorithm based on dihedral covers of knots.
result Explicit algorithm for all linking numbers.
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.
New CMC surfaces with dihedral symmetry constructed from Darboux transforms.
problem Constructing closed CMC surfaces with specific symmetries.
method Using Darboux transforms on multiple covers to create new CMC surfaces.
result Explicit parametrisations of new closed CMC surfaces with dihedral symmetry.
Sharp obstruction found for extending knot group quotients over surfaces in .
problem Determining when a knot group quotient extends over a surface in .
method Sharp obstruction using Seifert form and explicit construction when possible.
result Sharp obstruction and explicit construction of surfaces.
We consider knots equipped with a representation of their knot groups onto a dihedral group D_{2n} (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relati…
Character variety of a 6-punctured sphere mapped to CP^3 with specific branch points.
problem Characterizing the traceless SU(2) character variety of a 6-punctured 2-sphere.
method Analysis of 2-fold branched cover and application of Narasimhan-Ramanan theorem.
result Character variety corresponds to a 2-fold branched cover of CP^3 over a singular Kummer surface.
The study finds infinitely many twist knot complements with totally geodesic surfaces.
problem Finding infinitely many twist knot complements with a specific number of totally geodesic surfaces.
method Using a family of twist knot complements and their dihedral covers, the authors construct examples of hyperbolic 3-manifolds with totally geodesic surfaces.
result The construction of infinitely many non-commensurable hyperbolic 3-manifolds with exactly k totally geodesic surfaces for any positive integer k.
Consider the cyclic group C_2 of order two acting by complex-conjugation on the unit circle S^1. The main result is that a finitely dominated manifold W of dimension > 4 admits a cocompact, free, discontinuous action by the infinite dihedral group D_\infty if and only if W is the infinite cyclic cover of a free C_2-man…
Algorithms compute invariants of 4-manifolds as branched covers.
problem Computing invariants of 4-manifolds as branched covers.
method Diagrammatic algorithms using tri-plane diagrams and permutations.
result Automated algorithm for computing Kjuchukova's homotopy-ribbon obstruction.
Irreducibility of the space of dihedral covers of the projective line of a given numerical typemath.AG
We show in this paper that the set of irreducible components of the family of Galois coverings of P^1_C with Galois group isomorphic to D_n is in bijection with the set of possible numerical types. In this special case the numerical type is the equivalence class (for automorphisms of D_n) of the function which to each …
The paper describes a cover of strata of k-differentials with a formula for fiber cardinality.
problem Understanding the ramification locus and cardinality of fibers in strata of k-differentials.
method Intersection calculations on multi-scale compactification and flat geometry.
result A formula for the cardinality of each fiber involving the k-factorial function.
Smoothly approximating convex polytopes proves dihedral rigidity.
problem Proving dihedral rigidity for convex polytopes.
method Smooth approximation approach following Brendle.
result Proved Gromov's dihedral rigidity conjecture for convex polytopes.
New index theory proves Gromov's dihedral conjectures.
problem Comparisons and rigidity of scalar curvatures, mean curvatures, and dihedral angles.
method Developed a new index theory for manifolds with polyhedral boundary.
result Proved Gromov's dihedral extremality and rigidity conjectures.
Study dihedral spherical surfaces and their foliations.
problem Characterize dihedral spherical surfaces and their foliations.
method Define and analyze dihedral surfaces and their foliations, introduce geometric decompositions and deformations.
result Determine the dimension of the moduli space for dihedral surfaces.
Characterizes conical angles for metrics with dihedral symmetry.
problem Understanding metrics with specific symmetry properties.
method Using recent results on local invariants of quadratic differentials.
result Complete characterization of conical angles for dihedral spherical metrics.
The paper introduces two-tone colorings for links and shows conditions for surjective dihedral representations.
problem The challenge is to find conditions for links to admit surjective dihedral representations.
method The method involves introducing two-tone colorings and providing conditions for the link groups to admit such representations.
result Any link with at least 3 components admits a surjective homomorphism to the dihedral group of arbitrary degree.
We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since d…
The paper characterizes finite metacyclic subgroups in mapping class groups of surfaces.
problem Characterizing finite metacyclic subgroups in mapping class groups of surfaces.
method Derives necessary and sufficient conditions for conjugates of torsion elements to generate finite split non-abelian metacyclic subgroups.
result Complete classification of certain finite metacyclic subgroups in for .
The paper proves a spacetime version of dihedral rigidity for cubes in 3D spacetime.
problem Proving dihedral rigidity for cubic initial data sets in 3D spacetime.
method By studying the level sets of spacetime harmonic functions and extending previous work on dihedral rigidity for prisms in hyperbolic space.
result The paper proves dihedral rigidity for cubes in 3D spacetime, extending previous results.
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
Symmetric elastic knotsmath.DG
Symmetric elastic knots are found for certain classes with dihedral symmetry.
problem Finding elastic knots with specific symmetries.
method Minimizing bending energy under dihedral symmetry constraints.
result Existence of dihedral symmetric elastic knots, including a figure-eight union for the trefoil.
Study on geodesics and dihedral groups in lattices.
problem Growth and distribution of conjugacy classes of dihedral subgroups.
method Generalizing earlier work on reciprocal geodesics, proving equidistribution.
result Reciprocal geodesics are equidistributed in the unit tangent bundle.
Let K be a knot in an integral homology 3-sphere and let B denote the 2-fold branched cover of the integral homology sphere branched along K. We construct a map from the slice of characters with trace free along meridians in the SL(2, C)-character variety of the knot exterior to the SL(2, C)-character variety of 2-fold…
Study growth rates of specific hyperbolic Coxeter groups with fixed dihedral angles.
problem Arithmetic properties of growth rates in hyperbolic Coxeter groups.
method Analyzing arithmetic properties of growth rates for specific Coxeter groups with fixed dihedral angles.
result Growth rates are always Perron numbers.
Minimal surfaces with dihedral symmetry are studied as angles converge to zero.
problem Understanding minimal surfaces with dihedral symmetry as angles approach zero.
method Analyzing the limit of minimal surfaces in wedges with varying angles and using the implicit function theorem.
result New minimal surfaces are discovered and existence proofs are simplified.
Study of quandle coloring quivers with dihedral quandles.
problem Link invariants and their enhancements using quandles.
method Introduced shadow quandle coloring quivers and cocycle quivers, studied equivalence with quandle coloring numbers and shadow quandle cocycle invariants.
result Equivalence of quandle coloring quivers with quandle coloring numbers and shadow quandle cocycle quivers with shadow quandle cocycle invariants for specific dihedral quandles.
Compute Bredon homology for a specific type of Artin groups.
problem Calculate Bredon homology for Artin groups of dihedral type.
method Compute Bredon homology groups of the classifying space for virtually cyclic subgroups with K-theory coefficients.
result Computed Bredon homology groups for Artin groups of dihedral type.
Algebraic methods prove knot primality using Floer homology.
problem Proving the primality of knots.
method Knot Floer homology, metacyclic representations, and twisted homology.
result Primality tests have proven primality for over 99.6% of knots.
Classifies orbits of Hurwitz actions on dihedral quandles.
problem Classifying orbits of Hurwitz actions on dihedral quandles.
method Introduced three computable invariants to classify orbits.
result Complete classification of orbits under Hurwitz action.
Homotopy classification for certain 4-manifolds with dihedral fundamental groups.
problem Classifying the homotopy types of specific 4-manifolds with dihedral fundamental groups.
method Using quadratic 2-type and combining with results from Hambleton-Kreck and Bauer.
result Homotopy types of finite oriented Poincaré 4-complexes are determined by their quadratic 2-type when fundamental group is dihedral.
The study characterizes torus links' coloring quivers using dihedral quandles.
problem Characterizing the structure of coloring quivers for torus links.
method Exhaustively determining all possible numbers of colorings and their interconnections.
result The quiver structure varies based on the number of colorings.
In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.
The paper creates exotic 4-manifold structures with a specific group.
problem Producing exotic structures on 4-manifolds with infinite dihedral fundamental group.
method Using specific conditions on and , the paper constructs these structures.
result The existence of infinite exotic structures on 4-manifolds with infinite dihedral fundamental group.
The study sets limits on dihedral angles of large hyperbolic polyhedra.
problem Establishing bounds on dihedral angles of hyperbolic Coxeter polyhedra.
method Developed a constructive procedure for Coxeter polyhedra with prescribed dihedral angles.
result Classification of ADEG-polyhedra with specific dihedral angles and no disjoint facets.
Researchers found the Wigner derivative and its inverse are equal for spherical tetrahedra.
problem Computing the relationship between dihedral angles and edge lengths in tetrahedra.
method Computed the Wigner derivative and its inverse for spherical tetrahedra.
result The Wigner derivative and its inverse are equal for spherical tetrahedra.
The paper proves rigidity for submanifolds in warped product manifolds.
problem Rigidity of submanifolds in warped product manifolds.
method Dihedral extremality and rigidity theorem for submanifolds with polyhedral boundary.
result Dihedral rigidity results for hyperbolic polyhedra in flat warped product spaces.
Proves a conjecture about prism shapes with nonnegative curvature.
problem Understanding the rigidity of prism shapes with nonnegative curvature.
method Constructing free boundary minimal hypersurfaces and extending a dimension descent idea.
result Dihedral angles of certain prisms cannot exceed Euclidean models unless isometric.
New formula for signature of topological branched covers, extending classical results.
problem Computing signatures of topological branched covers with non-smooth points.
method Intersection Homology signature and homotopy ribbon obstructions.
result New formula for the contribution to signature from non-smooth points.
Study lengths of 3-cocycles for specific quandles, finding knot properties.
problem Determining lengths of 3-cocycles for 7-dihedral and octahedral quandles.
method Analyzing specific quandles to find lengths of 3-cocycles.
result 2-twist-spun -knot and 4-twist-spun trefoil have triple point number eight.
Indecomposable -complexesmath.GT
We show that if is an indecomposable -complex and XZ/2Z2m…
The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.
problem Understanding the combinatorial types of 3D polytopes in specific Riemannian manifolds.
method Analysis of mean curvature convex Riemannian polyhedra with non-obtuse dihedral angles in positive scalar curvature 3-manifolds.
result Determination of combinatorial types of 3D simple convex polytopes.
We prove that, both in the hyperbolic and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study properties of such polyhedral surfaces. In particular, we prove that the volume …