Cyclic coverings of virtual link diagrams

A virtual link diagram is called mod [Formula: see text] almost classical if it admits an Alexander numbering valued in integers modulo [Formula: see text], and a virtual link is called mod [Formula: see text] almost classical if it has a mod [Formula: see text] almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod [Formula: see text] almost classical virtual link diagram from a given virtual link diagram, which we call an [Formula: see text]-fold cyclic covering diagram. The main result is that [Formula: see text]-fold cyclic covering diagrams obtained from two equivalent virtual link diagrams are equivalent. Thus, we have a well-defined map from the set of virtual links to the set of mod [Formula: see text] almost classical virtual links. Some applications are also given.

Multivariate Alexander colorings

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module [Formula: see text] over the Laurent polynomial ring [Formula: see text]. If [Formula: see text] is a diagram of a link [Formula: see text] with [Formula: see text] components, then the colorings of [Formula: see text] with values in [Formula: see text] form a [Formula: see text]-module [Formula: see text]. Extending a result of Inoue [Knot quandles and infinite cyclic covering spaces, Kodai Math. J. 33 (2010) 116–122], we show that [Formula: see text] is isomorphic to the module of [Formula: see text]-linear maps from the Alexander module of [Formula: see text] to [Formula: see text]. In particular, suppose [Formula: see text] is a field and [Formula: see text] is a homomorphism of rings with unity. Then [Formula: see text] defines a [Formula: see text]-module structure on [Formula: see text], which we denote [Formula: see text]. We show that the dimension of [Formula: see text] as a vector space over [Formula: see text] is determined by the images under [Formula: see text] of the elementary ideals of [Formula: see text]. This result applies in the special case of Fox tricolorings, which correspond to [Formula: see text] and [Formula: see text]. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine [Formula: see text]; this observation corrects erroneous statements of Inoue [Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications 10 (2001) 813–821; op. cit.].

Variation of the Variety of Minimal Rational Tangents of Cyclic Coverings

Let π: X → ℙn be the d-cyclic covering branched along a smooth hypersurface Y ⊂ ℙn of degree d, 3 ≤ d ≤ n. We identify the minimal rational curves on X with d-tangent lines of Y and describe the scheme structure of the variety of minimal rational tangents 𝒞x ⊂ ℙTx(X) at a general point x ∈ X. We also show that the projective isomorphism type of 𝒞x varies in a maximal way as x moves over general points of X.

An explicit relation between knot groups in lens spaces and those in S3

For a cyclic covering map [Formula: see text] between two pairs of a 3-manifold and a knot each, we describe the fundamental group [Formula: see text] in terms of [Formula: see text]. As a consequence, we give an alternative proof for the fact that certain knots in [Formula: see text] cannot be represented as the preimage of any knot in a lens space, which is related to free periods of knots. In our proofs, the subgroup of a group [Formula: see text] generated by the commutators and the [Formula: see text]th power of each element of [Formula: see text] plays a key role.

Cyclic covering of a module over an Artinian ring

Given a commutative ring with identity [Formula: see text] and an [Formula: see text]-module [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a cyclic covering of [Formula: see text], if this module is the union of the cyclic submodules [Formula: see text], where [Formula: see text]. Such covering is said to be irredundant, if no proper subset of [Formula: see text] is a cyclic covering of [Formula: see text]. In this work, an irredundant cyclic covering of [Formula: see text] is constructed for every Artinian commutative ring [Formula: see text]. As a consequence, a cyclic covering of minimal cardinality of [Formula: see text] is obtained for every finite commutative ring [Formula: see text], extending previous results in the literature.

The p-rank of the reduction mod p of Jacobians and Jacobi sums

Let YK → XK be a ramified cyclic covering of curves, where K is a cyclotomic field. In this work we study the p-rank of the reduction mod p of a model of the Jacobian of YK. In this way, we obtain counterparts of the Deuring polynomial, defined for elliptic curves, for genus greater than one. We provide a new point of view of this subject in terms of L-functions. To carry out this study we use the relationship between Jacobi sums and L-functions. This is established in [A. Weil, Jacobi sums as "Grössencharaktere", Trans. Amer. Math. Soc. 73 (1952) 487–495] for the case of Fermat curves. We also give a new proof of a result of Deligne concerning the constant terms of these L-functions and Jacobi sums.

ORDERS OF BRANCHED COVERINGS OF LINKS

Let M, M′ be compact oriented 3-manifolds and L′ a link in M′. We prove that, under certain conditions, the topological types of M and (M′, L′) determine the degree of a cyclic covering p : M → M′, branched over L′.