arXiv research

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

We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection wit…

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…

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

We prove that the Waldhausen Nil-group associated to a virtually cyclic groups that surjects onto the infinite dihedral group vanishes if and only if the corresponding Farrell Nil-group associated to the canonical index two subgroup is trivial. The proof uses the transfer map to establish one direction, and uses contro…

The symplectic representation of mapping classes is not surjective for certain types of mapping classes.

The paper explores representations of specific knot groups and their properties.

Classifies symmetries of knots using group actions and orthogonal representation theory.

New properties established for SO(3) quantum representations, showing density and surjectivity.

The paper calculates Alexander polynomials for knots using finite group representations.

We introduce a version of Khovanov homology for alternating links with marking data, $ω$, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in \cite{KM_unknot} for this marked Khovanov homology collapses on the $E_2$ page fo…

We study the representation spaces $R(K;\bf{i})$ as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots $P(p,q,r)$ with $p, q, r$ pairwise coprime, these appear to be non-degenerate and comprise representations in SU(2) that are not b…

Given an ideal triangulation of a connected 3-manifold with non-empty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject to Thurston's gluing equations. From this, one can recover a representation of th…

New index theory proves Gromov's dihedral conjectures.

The objective of this paper is to determine the finite dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to those of the infinite dihedral group, we give the algebra the name iD-infinity.

Study dihedral spherical surfaces and their foliations.

Characterizes conical angles for metrics with dihedral symmetry.

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.

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…

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.

We define and study the set ${\mathcal E}(ρ)$ of end invariants of a $\SL(2,C)$ character $ρ$ of the one-holed torus $T$. We show that the set ${\mathcal E}(ρ)$ is the entire projective lamination space $\mathscr{PL}$ of $T$ if and only if (i) $ρ$ corresponds to the dihedral representation, or (ii) $ρ$ is real and corr…

Study links using Soergel bimodules and Serre duality.

The pair (K,r) consisting of a knot K and a surjective map r from the knot group onto a dihedral group is said to be a p-colored knot. D. Moskovich conjectured that for any odd prime p there are exactly p equivalence classes of p-colored knots up to surgery along unknots in the kernel of the coloring. We show that ther…

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

Compute Bredon homology for a specific type of Artin groups.

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…

Study of quandle coloring quivers with dihedral quandles.

Dihedral linking invariant uses knot colorings to distinguish knots.

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n $\in$ N for which there exists a surjection between the n-and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, …

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.

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.

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.

SSNL improves simulation-based inference for high-dimensional data.

We describe a collection of graded rings which surject onto Webster rings for sl(2) and which should be related to certain categories of singular Soergel bimodules. In the first non-trivial case, we construct a categorical braid group action which categorifies the Burau representation.

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 study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral h…