Cyclic Coverings Research Articles

Invariant rings of the special orthogonal group have nonunimodal h-vectors

For [Formula: see text] an infinite field of characteristic other than two, consider the action of the special orthogonal group [Formula: see text] on a polynomial ring via copies of the regular representation. When [Formula: see text] has characteristic zero, Boutot’s theorem implies that the invariant ring has rational singularities; when [Formula: see text] has positive characteristic, the invariant ring is [Formula: see text]-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for [Formula: see text] as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example, it readily yields the [Formula: see text]-invariant and information on the Hilbert series. Indeed, we use this to show that the [Formula: see text]-vector of the invariant ring for [Formula: see text] need not be unimodal.

Cyclic coverings of the 3-sphere branched over wild knots of dynamically defined type

Let [Formula: see text] be a tame knot and consider an [Formula: see text] beaded necklace [Formula: see text] which is the union of [Formula: see text] consecutive disjoint closed round balls (pearls) [Formula: see text], [Formula: see text]. An [Formula: see text] pearl chain necklace [Formula: see text] is the union of [Formula: see text] and [Formula: see text]. We will construct, via the action of a Kleinian group, a sequence of nested pearl chain necklaces [Formula: see text] whose inverse limit is a wild knot of dynamically defined type [Formula: see text]. In this paper, we will prove some topological properties of this kind of wild knots; in particular, we generalize the construction of cyclic branched coverings for this case, and we show that there exists a wild knot of dynamically defined type such that [Formula: see text] is an [Formula: see text]-fold cyclic branched covering of [Formula: see text] along it, for [Formula: see text].

The classifying element for quotients of Fermat curves

Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞ . Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q .

Knot concordance invariants from Seiberg–Witten theory and slice genus bounds in 4-manifolds

We construct a new family of knot concordance invariants [Formula: see text], where [Formula: see text] is a prime number. Our invariants are obtained from the equivariant Seiberg–Witten–Floer cohomology, constructed by the author and Hekmati, applied to the degree [Formula: see text] cyclic cover of [Formula: see text] branched over [Formula: see text]. In the case [Formula: see text], our invariant [Formula: see text] shares many similarities with the knot Floer homology invariant [Formula: see text] defined by Hom and Wu. Our invariants [Formula: see text] give lower bounds on the genus of any smooth, properly embedded, homologically trivial surface bounding [Formula: see text] in a definite [Formula: see text]-manifold with boundary [Formula: see text].

The Moduli Space of Cyclic Covers in Positive Characteristic

Abstract We study the $p$-rank stratification of the moduli space $\operatorname{\mathcal{A}\mathcal{S}\mathcal{W}}_{(d_{1},d_{2},\ldots ,d_{n})}$, which represents $\mathbb{Z}/p^{n}$-covers in characteristic $p>0$ whose $\mathbb{Z}/p^{i}$-subcovers have conductor $d_{i}$. In particular, we identify the irreducible components of the moduli space and determine their dimensions. To achieve this, we analyze the ramification data of the represented curves and use it to classify all the irreducible components of the space. In addition, we provide a comprehensive list of pairs $(p,(d_{1},d_{2},\ldots ,d_{n}))$ for which $\operatorname{\mathcal{A}\mathcal{S}\mathcal{W}}_{(d_{1},d_{2},\ldots ,d_{n})}$ in characteristic $p$ is irreducible. Finally, we investigate the geometry of $\operatorname{\mathcal{A}\mathcal{S}\mathcal{W}}_{(d_{1},d_{2},\ldots ,d_{n})}$ by studying the deformations of cyclic covers that vary the $p$-rank and the number of branch points.

Application of gahnite-reinforced cyclic olefin copolymer cover sheets on Si solar cell for augmenting the power conversion efficiency

Application of gahnite-reinforced cyclic olefin copolymer cover sheets on Si solar cell for augmenting the power conversion efficiency

Cyclic coverings of genus curves of Sophie Germain type

Abstract We consider cyclic unramified coverings of degree d of irreducible complex smooth genus $2$ curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order d. The rich geometry of the associated Prym map has been studied in several papers, and the cases $d=2, 3, 5, 7$ are quite well understood. Nevertheless, very little is known for higher values of d. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for $d\ge 11$ prime such that $\frac {d-1}2$ is also prime. We use results of arithmetic nature on $GL_2$ -type abelian varieties combined with theta-duality techniques.

Skeletal filtrations of the fundamental group of a non-archimedean curve

In this paper we study skeleta of residually tame coverings of a marked curve over a non-archimedean field. We first prove a simultaneous semistable reduction theorem for residually tame coverings, which we then use to construct a tropicalization functor from the category of residually tame coverings of a marked curve (X,D) to the category of tame coverings of a metrized complex Σ associated to (X,D). We enhance the latter category by adding a set of gluing data to every covering and we show that this yields an equivalence of categories. We use this skeletal interpretation to define the absolute decomposition and inertia group of a curve, which can be seen as the first subgroups in a ramification filtration of the fundamental group of the curve. We prove that the cyclic coverings that arise from the corresponding decomposition and inertia quotients coincide with the coverings that arise from the toric and connected parts of the analytic Jacobian of the curve.

The Novel Generally Described Graphs for Cyclic Orthogonal Double Covers of Some Circulants

The Novel Generally Described Graphs for Cyclic Orthogonal Double Covers of Some Circulants

Weierstrass Semigroups from Cyclic Covers of Hyperelliptic Curves

Weierstrass Semigroups from Cyclic Covers of Hyperelliptic Curves

A polyhedral approach to the arithmetic and geometry of hyperbolic link complements

We provide an elementary polyhedral approach to study and deduce results about the arithmeticity and commensurability of an infinite family of hyperbolic link complements [Formula: see text] for [Formula: see text]. The manifold [Formula: see text] is the complement of [Formula: see text] by the ([Formula: see text])-link chain [Formula: see text] and has [Formula: see text] cusps. We show that [Formula: see text] is closely related to a hyperbolic Coxeter orbifold that is commensurable to an orbifold with a single cusp. Vinberg’s arithmeticity criterion and certain cusp density and volume computations allow us to reproduce some of the main results in [20] and [18] about [Formula: see text] in a comparatively elementary and direct way. As a by-product, we give a rigorous proof of Thurston’s volume formula for [Formula: see text] and deduce that, for [Formula: see text], the volume of [Formula: see text] is strictly bigger than the volume of the ([Formula: see text])-cyclic cover over one component of the Whitehead link.

Some evaluations of the Jones polynomial for certain families of weaving knots

In this paper, we derive formulae for the determinant of weaving knots W(3,n) and W(p,2). We calculate the dimension of the first homology group with coefficients in Z3 of the double cyclic cover of the 3-sphere S3 branched over W(3,n) and W(p,2) respectively. As a consequence, we obtain a lower bound of the unknotting number of W(3,n) for certain values of n.

Infinitesimal rigidity for cubulated manifolds

We prove the infinitesimal rigidity of some geometrically infinite hyperbolic 4- and 5-manifolds. These examples arise as infinite cyclic coverings of finite-volume hyperbolic manifolds obtained by colouring right-angled polytopes, already described in the papers (Battista in Trans Am Math Soc 375(04):2597–2625, 2022; Italiano et al. in Invent Math, 2022. https://doi.org/10.1007/s00222-022-01141-w). The 5-dimensional example is diffeomorphic to Ntimes {{mathbb {R}}} for some aspherical 4-manifold N which does not admit any hyperbolic structure. To this purpose, we develop a general strategy to study the infinitesimal rigidity of cyclic coverings of manifolds obtained by colouring right-angled polytopes.

Cyclic coverings of graphs. Counting rooted spanning forests and trees, Kirchhoff index, and Jacobians

The purpose of this survey is to describe invariants of cyclic coverings of graphs. The covered graph is assumed to be fixed, and the cyclic covering group has an arbitrarily large order. A classical example of such a covering is a circulant graph. It covers a one-vertex graph with a prescribed number of loops. More sophisticated objects representing the family of cyclic coverings include $I$-, $Y$-, and $H$-graphs, generalized Petersen graphs, sandwich-graphs, discrete tori, and many others. We present analytic formulae for counting rooted spanning forests and trees in cyclic coverings, establish their asymptotics, and study the arithmetic properties of these quantities. Moreover, in the case of circulant graphs we give exact formulae for computing the Kirchhoff index and present structural theorems for the Jacobians of such graphs. Bibliography: 95 titles.

Циклические накрытия графов. Перечисление отмеченных остовных лесов и деревьев, индекс Кирхгофа и якобианы

Цель настоящего обзора - изучение инвариантов циклических накрытий графов. При этом накрываемый граф предполагается фиксированным, а циклическая группа накрытия имеет сколь угодно большой порядок. Классическим примером такого накрытия является циркулянтный граф. Он накрывает одновершинный граф с заданным числом петель. Более сложными представителями семейства циклических накрытий являются $I$-, $Y$-, $H$-графы, обобщенные графы Петерсена, сэндвич-графы, дискретные торы и многие другие. В обзоре приведены аналитические формулы, позволяющие вычислять число отмеченных остовных лесов и деревьев в циклических накрытиях, найдена их асимптотика и изучены арифметические свойства этих чисел. Кроме того, для циркулянтных графов указаны точные формулы для вычисления индекса Кирхгофа и приведены структурные теоремы о строении якобианов таких графов. Библиография: 95 названий.

Geometric generalizations of the square sieve, with an application to cyclic covers

AbstractWe formulate a general problem: Given projective schemes and over a global field K and a K‐morphism η from to of finite degree, how many points in of height at most B have a pre‐image under η in ? This problem is inspired by a well‐known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a nontrivial answer to the general problem when and is a prime degree cyclic cover of . Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.

Toward the group completion of the Burau representation

Following Boardman-Vogt, McDuff, Segal, and others, we construct a monoidal topological groupoid or space of finite subsets of the plane, and interpret the Burau representation of knot theory as a topological quantum field theory defined on it. Its determinant or writhe is an invertible braided monoidal TQFT which group completes to define a Hopkins-Mahowald model for integral homology as an E 2 E_2 Thom spectrum. We use these ideas to construct an infinite cyclic (Alexander) cover for the space of finite subsets of C \mathbb {C} , and we argue that the TQFT defined by Burau is closely related to the SU(2)-valued Wess-Zumino-Witten model for string theory on R + 3 \mathbb {R}^3_+ .

Cyclic and Abelian coverings of real varieties

We describe the birational and the biregular theory of cyclic and Abelian coverings between real varieties.

K-theoretic torsion and the zeta function

We generalize to higher algebraic $K$-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion.

Orthogonal Labeling for Some Different Infinite Graph Classes

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the mid 1960s. Graph labeling is one of the fascinating areas of graph theory with wide ranging applications. A collection $\mathcal{A}$ of isomorphic copies of a given subgraph $G$ of $T$ is said to be an orthogonal double cover (ODC) of the graph $T$ by $G$, if every edge of $T$ belongs to exactly two members of $\mathcal{A}$ and any two different elements from $\mathcal{A}$ share at most one edge. The existence of an orthogonal labeling of a graph $G$ with respect to a certain group implies the existence of the cyclic orthogonal double cover of the circulant graphs on that group. In this paper, we prove the existence of orthogonal labeling for some different infinite graph classes and hence, the existence of the cyclic orthogonal double cover of some different infinite circulant graphs.