arXiv research

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.

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…

Dihedral linking invariant uses knot colorings to distinguish knots.

Consider a dihedral cover $f: Y\to X$ with $X$ and $Y$ four-manifolds and $f$ branched along an oriented surface embedded in $X$ with isolated cone singularities. We prove that only a slice knot can arise as the unique singularity on an irregular dihedral cover $f: Y\to S^4$ if $Y$ is homotopy equivalent to $\mathbb{CP…

The paper details folding of branched covers of the 3-sphere over knots.

Let $K\subset S^3$ be a Fox $p$-colored knot and assume $K$ bounds a locally flat surface $S\subset B^4$ over which the given $p$-coloring extends. This coloring of $S$ induces a dihedral branched cover $X\to S^4$. Its branching set is a closed surface embedded in $S^4$ locally flatly away from one singularity whose li…

New CMC surfaces with dihedral symmetry constructed from Darboux transforms.

Sharp obstruction found for extending knot group quotients over surfaces in $B^4$.

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…

Let $M$ be a connected, closed, oriented three-manifold and $K$, $L$ two rationally null-homologous oriented simple closed curves in $M$. We give an explicit algorithm for computing the linking number between $K$ and $L$ in terms of a presentation of $M$ as an irregular dihedral $3$-fold cover of $S^3$ branched along a…

Kjuchukova's $Ξ_p$ invariant gives a ribbon obstruction for Fox $p$-colored knots. The invariant is derived from dihedral branched covers of 4-manifolds, and is needed to calculate the signatures of these covers, when singularities on the branching sets are present. In this note, we give an algorithm for evaluating $Ξ_…

The study finds infinitely many twist knot complements with totally geodesic surfaces.

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.

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.

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.

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.

Smooth approximations bound dihedral angles of convex polytopes.

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

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

Study on geodesics and dihedral groups in lattices.

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…

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.

We exhibit the traceless $SU(2)$ character variety of a 6-punctured 2-sphere as a 2-fold branched cover of ${\mathbb{C}}P^3$, branched over the singular Kummer surface, with the branch locus in $R(S^2,6)$ corresponding to the binary dihedral representations. This follows from an analysis of the map induced on $SU(2)$ c…

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.

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

Algebraic methods prove knot primality using Floer homology.

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.

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

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 …

Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. Classically, if $B$ is smoothly embedded in $Y$, the signature $σ(X)$ can be computed from data about $Y$, $B$ and the local degrees of $f$. When $f$ is an irregular dihedral cover and $B\subset Y$ smoothly embedded away fr…

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 if $X$ is an indecomposable $PD_3$-complex and $π_1(X) is the fundamental group of a reduced finite graph of finite groups but is not virtually cyclic then$X$is orientable, the underlying graph is a tree, all the edge groups are$Z/2Z$and all but at most one of the vertex groups is dihedral of order$2m…

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…