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 A remark on branched cyclic covers

1993 ◽  
Vol 87 (3) ◽  
pp. 237-240
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Cyclic Covers ◽  
Branched Cyclic Covers

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.

RECURSION FORMULAS FOR SOME ABELIAN KNOT INVARIANTS

2000 ◽  
Vol 09 (03) ◽  
pp. 413-422 ◽  
Author(s):  
WAYNE H. STEVENS
Keyword(s):  
Recursion Formula ◽  
Alexander Polynomial ◽  
Square Roots ◽  
Knot Invariants ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Recursion Formulas ◽  
Branched Cyclic Covers ◽  
Linear Recursion ◽  
Fold Cover

Let K be a tame knot in S3. We show that the sequence of cyclic resultants of the Alexander polynomial of K satisfies a linear recursion formula with integral coefficients. This means that the orders of the first homology groups of the branched cyclic covers of K can be computed recursively. We further establish the existence of a recursion formula that generates sequences which contain the square roots of the orders for the odd-fold covers that contain the square roots of the orders for the even-fold covers quotiented by the order for the two-fold cover. (That these square roots are all integers follows from a theorem of Plans.)

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 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.

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