arXiv research

The dihedral genus of a knot is related to its signature and minimal surface genus.

We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry group the natural $\bfZ_2$ extension of the dihedral group $D_n$. The surfaces are …

The paper constructs surfaces of high genus with three ends.

Unique genus 2 minimal surface in S^3 with specific symmetry group.

The study constructs minimal surfaces in a product space with specific properties.

For fixed large genus, we construct families of complete immersed minimal surfaces in R3 with four ends and dihedral symmetries. The families exist for all large genus and at an appropriate scale degenerate to the plane.

We construct new constant mean curvature surfaces in H2xR. They arise as sister surfaces of Plateau solutions. It is a family of MC 1/2 surfaces with k ends, genus 1 and k-fold dihedral symmetry, k greater 2. The surfaces are Alexandrov- embedded.

We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square.

The article explores symmetric maps on surfaces, focusing on semi-equivelar maps.

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits corr…

Moebius-Kantor graph connects multiple groups and topological properties.

With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space $H^3$, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface has an ess…

New index theory proves Gromov's dihedral conjectures.

Study dihedral spherical surfaces and their foliations.

Characterizes conical angles for metrics with dihedral symmetry.

The paper introduces two-tone colorings for links and shows conditions for surjective dihedral representations.

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.

The paper proves a spacetime version of dihedral rigidity for cubes in 3D spacetime.

Smooth approximations bound dihedral angles of convex polytopes.

Symmetric elastic knots are found for certain classes with dihedral symmetry.

Study on geodesics and dihedral groups in lattices.

The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable 3-dimensional handlebody of genus g can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g-1)/|G|, u…

Minimal surfaces with dihedral symmetry are studied as angles converge to zero.

Compute Bredon homology for a specific type of Artin groups.

Study of quandle coloring quivers with dihedral quandles.

Dihedral linking invariant uses knot colorings to distinguish knots.

Classifies orbits of Hurwitz actions on dihedral quandles.

Homotopy classification for certain 4-manifolds with dihedral fundamental groups.

The study characterizes torus links' coloring quivers using dihedral quandles.

We construct many closed, embedded mean curvature self-shrinking surfaces $Σ_g^2\subseteq\mathbb{R}^3$ of high genus $g=2k$, $k\in \mathbb{N}$. Each of these shrinking solitons has isometry group equal to the dihedral group on $2g$ elements, and comes from the "gluing", i.e. desingularizing of the singular union, of th…

In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.

Minimal surfaces span periodic curves in 3D space.

The paper creates exotic 4-manifold structures with a specific group.

The study sets limits on dihedral angles of large hyperbolic polyhedra.

Researchers found the Wigner derivative and its inverse are equal for spherical tetrahedra.

The paper proves rigidity for submanifolds in warped product manifolds.

Proves a conjecture about prism shapes with nonnegative curvature.

Given a closed oriented PL four-manifold $X$ and a closed surface $B$ embedded in $X$ with isolated cone singularities, we give a formula for the signature of an irregular dihedral cover of $X$ branched along $B$. For $X$ simply-connected, we deduce a necessary condition on the intersection form of a simply-connected i…

Study lengths of 3-cocycles for specific quandles, finding knot properties.

New methods compare Steklov eigenspaces of free boundary minimal surfaces in balls.

We obtain compact orientable embedded surfaces with constant mean curvature $0<H<\frac{1}{2}$ and arbitrary genus in $\mathbb{S}^2\times\mathbb{R}$. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature $\frac{1}{2}$ tangent along an equator. This is a particular case of a conjug…

The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.

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 …

We give an original analytic construction of hyperkahler ALF metrics on some ALE spaces of dihedral type, namely the spaces corresponding to minimal resolutions of Kleinian quotients relative to some binary dihedral group.

We show that a finite dihedral group does not act pseudofreely and locally linearly on a 2k-dimensional sphere, if k > 1. This answers a question of R. S. Kulkarni from 1982.

Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. For the smallest example of order 6 that is an ex…

We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W, replacing the group von Neumann algebra NW with the Hecke-von Neumann algebra N_qW.

A quandle coloring obstruction prevents a specific link from being ribbon concordant.