The number of real ovals of a cyclic cover of the sphere

2016 ◽  
Vol 145 (6) ◽  
pp. 2639-2647
Author(s):  
Francisco-Javier Cirre ◽  
Peter Turbek
Keyword(s):  
Cyclic Cover

Homology of Branched Coverings of 3-Manifolds

1992 ◽  
Vol 44 (1) ◽  
pp. 119-134
Author(s):  
John Hempel
Keyword(s):  
Classical Result ◽  
Quantitative Information ◽  
Cyclic Cover ◽  
Branched Covers ◽  
Branched Coverings ◽  
Homology Groups

AbstractWe give a relation between the homology groups H1() and H1 (M) for a branched cyclic cover → M of arbitrary closed, oriented 3-manifolds which generalizes a classical result of Plans on covers of S3 branched over a knot and provides other quantitative information as well. We include a general "free calculus" procedure for computing homology groups of branched covers and reinterpret the results in this computational setting.

scholarly journals FINITE COVERS OF THE INFINITE CYCLIC COVER OF A KNOT

2009 ◽  
Vol 18 (01) ◽  
pp. 75-85
Author(s):  
J. O. BUTTON
Keyword(s):  
Finite Index ◽  
Commutator Subgroup ◽  
Surjective Homomorphism ◽  
Cyclic Cover ◽  
Finite Covers ◽  
Infinite Cyclic Cover ◽  
Simply Connected

We show that the commutator subgroup G′ of a classical knot group G need not have subgroups of every finite index, but it will if G′ has a surjective homomorphism to the integers and we give an exact criterion for that to happen. We also give an example of a knotted Sn in Sn+2 for all n ≥ 2 whose infinite cyclic cover is not simply connected but has no proper finite covers.

Knots which are branched cyclic covers of only finitely many knots

1985 ◽  
Vol 98 (2) ◽  
pp. 301-304
Author(s):  
Paul Strickland
Keyword(s):  
Cyclic Cover ◽  
Cyclic Covers ◽  
Infinite Cyclic Cover ◽  
Branched Cyclic Covers

In [5] we proved two results: theorem 1, which said that if k was a simple (2q – 1)-knot, q 1, then it was equivalent to the m-fold branched cyclic cover of another knot if and only if there existed an isometry u of its Blanchfield pairing 〈,〉, whose mth power was the map induced by a generator t of the group of covering translations associated with the infinite cyclic cover of k; and theorem 2, which showed that if k were the m-fold b.c.c. of two such knots, then these would be equivalent if and only if the corresponding isometries were conjugate by an isometry of 〈,〉. Using this second result, we present two cases where k may only be the m-fold b.c.c. of finitely many knots.

scholarly journals Simple cyclic covers of the plane and the Seshadri constants of some general hypersurfaces in weighted projective space

2021 ◽  
Author(s):  
Alex Küronya ◽  
Sönke Rollenske
Keyword(s):  
Projective Space ◽  
Irrational Number ◽  
Weighted Projective Space ◽  
Main Step ◽  
General Point ◽  
Cyclic Cover ◽  
Seshadri Constants ◽  
Cyclic Covers ◽  
Seshadri Constant

AbstractLet $$X \subset {\mathbb P}(1,1,1,m)$$ X ⊂ P ( 1 , 1 , 1 , m ) be a general hypersurface of degree md for some for $$d\ge 2$$ d ≥ 2 and $$m\ge 3$$ m ≥ 3 . We prove that the Seshadri constant $$\varepsilon ( {\mathcal O}_X(1), x)$$ ε ( O X ( 1 ) , x ) at a general point $$x\in X$$ x ∈ X lies in the interval $$\left[ \sqrt{d}- \frac{d}{m}, \sqrt{d}\right] $$ d - d m , d and thus approaches the possibly irrational number $$\sqrt{d}$$ d as m grows. The main step is a detailed study of the case where X is a simple cyclic cover of the plane.

scholarly journals The group of units on an affine variety

2014 ◽  
Vol 13 (08) ◽  
pp. 1450065 ◽  
Author(s):  
Timothy J. Ford
Keyword(s):  
Hyperelliptic Curve ◽  
Sufficient Conditions ◽  
Galois Cohomology ◽  
Closed Field ◽  
Affine Variety ◽  
Cyclic Cover ◽  
Group Of Units ◽  
Object Of Study ◽  
General Affine ◽  
Projective Completion

The object of study is the group of units 𝒪*(X) in the coordinate ring of a normal affine variety X over an algebraically closed field k. Methods of Galois cohomology are applied to those varieties that can be presented as a finite cyclic cover of a rational variety. On a cyclic cover X → 𝔸m of affine m-space over k such that the ramification divisor is irreducible and the degree is prime, it is shown that 𝒪*(X) is equal to k*, the non-zero scalars. The same conclusion holds, if X is a sufficiently general affine hyperelliptic curve. If X has a projective completion such that the divisor at infinity has r components, then sufficient conditions are given for 𝒪*(X)/k* to be isomorphic to ℤ(r-1).

scholarly journals Hodge Theory of Cyclic Covers Branched over a Union of Hyperplanes

2014 ◽  
Vol 66 (3) ◽  
pp. 505-524 ◽  
Author(s):  
Donu Arapura
Keyword(s):  
Projective Space ◽  
Prime Number ◽  
Hyperplane Arrangement ◽  
Hodge Theory ◽  
Cyclic Cover ◽  
Hodge Conjecture ◽  
Cyclic Covers ◽  
Generalized Hodge Conjecture ◽  
Ramification Locus

AbstractSuppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D and that U is the complement of the ramification locus in Y. The first theorem in this paper implies that the Beilinson-Hodge conjecture holds for U if certain multiplicities of D are coprime to the degree of the cover. For instance, this applies when D is reduced with normal crossings. The second theorem shows that when D has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of Y. The last section contains some partial extensions to more general nonabelian covers.

scholarly journals INVARIANTS FOR LEGENDRIAN KNOTS IN LENS SPACES

2011 ◽  
Vol 13 (01) ◽  
pp. 91-121 ◽  
Author(s):  
JOAN E. LICATA
Keyword(s):  
Cyclic Group ◽  
Group Action ◽  
Riemann Surfaces ◽  
Lens Spaces ◽  
Graded Algebra ◽  
Cyclic Cover ◽  
Legendrian Knots ◽  
Differential Graded Algebra ◽  
Cyclic Group Action

In this paper, we define invariants for primitive Legendrian knots in lens spaces L(p, q), q ≠ 1. The main invariant is a differential graded algebra [Formula: see text] which is computed from a labeled Lagrangian projection of the pair (L(p, q), K). This invariant is formally similar to a DGA defined by Sabloff which is an invariant for Legendrian knots in smooth S1-bundles over Riemann surfaces. The second invariant defined for K ⊂ L(p, q) takes the form of a DGA enhanced with a free cyclic group action and can be computed from a cyclic cover of the pair (L(p, q), K).

Sign in / Sign up

Export Citation Format

Share Document