arXiv research

The study counts critical points in knot cobordisms using abelian and metacyclic invariants.

The paper solves conditions for metacyclic actions on surfaces, including upper bounds and subgroup classifications.

Algebraic methods prove knot primality using Floer homology.

The paper characterizes finite metacyclic subgroups in mapping class groups of surfaces.

A {\em solvable} cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of l…

The paper explores infinite metacyclic subgroups in mapping class groups of surfaces.

Let p be an odd prime and D_p a dihedral group of order 2p. Let ρ: G(K) --> D_p --> GL(p,Z) be a non-abelian representation of the knot group G(K) of a knot K in 3-sphere. Let Δ_{ρ,K} (t) be the twisted Alexander polynomial of K associated to ρ. Let H(p) is the set of 2-bridge knots K, such that G(K) is mapped onto a n…

The study examines Lee metrics on groups and their properties.

Let $p$ be an odd prime. We construct a non-abelian extension $Γ$ of $S^1$ by $Z/p \times Z/p$, and prove that any finite subgroup of $Γ$ acts freely and smoothly on $S^{2p-1} \times S^{2p-1}$. In particular, for each odd prime $p$ we obtain free smooth actions of infinitely many non-metacyclic rank two $p$-groups on $…

A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We…

Conditions for Baumslag-Solitar subgroups in mapping class groups.

Study covers of surfaces, showing types and properties.

After showing that a covering space of surface bundles over $S^1$ factors as a `covering of fibers' followed by a `power covering', we prove that, for torus bundles, power coverings do not lower Heegaard genus, and that fiber coverings lower the genus only in special cases.

In "Rips complexes and covers in the uniform category" \cite{Rips} the authors define, following James \cite{J}, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given…

A foliation of a manifold M is called R-covered if its lift to the universal cover of M has space of leaves R. We show that there are many graph manifolds which admit taut foliations, but which do not admit any R-covered foliations. On the other hand, we show that these manifolds all have finite covers admitting R-cove…

The paper studies which branched covers can be lifted to braided embeddings.

We discuss construction of coverings of the unit ball of a finite dimensional Banach space. The well known technique of comparing volumes gives upper and lower bounds on covering numbers. This technique does not provide a construction of good coverings. Here we apply incoherent dictionaries for construction of good cov…

The paper details folding of branched covers of the 3-sphere over knots.

We study tori which are cyclic covers of the standard torus, that is, the deck transformation group of the covering map is cyclic. These covering tori can be parametrized in a natural way and we show that being cyclic is equivalent to certain arithmetic condition on these parameters. There is a natural $\mathrm{SL}(2,\…

Course on knots using branched coverings.

We introduce a framework for coverings of noncommutative spaces. Moreover, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.

Second part of a series on higher coverings of racks and quandles.

In this paper we consider completed coverings that are branched coverings in the sense of Fox. For completed coverings between PL manifolds we give a characterization of the existence of a monodromy representation and the existence of a locally compact monodromy representation. These results stem from a characterizatio…

Gromov-Thurston covers have Betti numbers as expected.

Formula compares metrics on branched coverings of line bundles.

Study how bottom of spectra changes with Riemannian coverings.

There are theories of coverings of $C^*$-algebras which can be included into a following list: coverings of commutative $C^*$-algebras, coverings of $C^*$-algebras of groupoids and foliations, coverings of noncommutative tori, the double covering of the quantum group $SO_q(3)$. This work is devoted to a single general …

Study of lifting maps in branched covers of 3-manifolds, showing non-injectivity.

Super tau-covers extend bihamiltonian hierarchies' symmetries.

A branched covering surface-knot is a surface-knot in the form of a branched covering over a surface-knot. For a branched covering surface-knot, we have a numerical invariant called the simplifying number. We show that branched covering surface-knots with degree three have the simplifying numbers less than three.

Paper defines embolic volume and relates it to Betti number using the covering trick.

Given a finite cover f:tilde{G} \to G and an embedding of tilde{G} in the plane, Negami conjectures that G embeds in P^2. Negami proved this conjecture for regular covers. In this paper we define two properties (Propserties V and E), depending on the cover tilde{G} and its embedding into S^2, and generalize Negami's re…

Study on spectral stability of Riemannian coverings.

Study involutions on 3D small covers, proving quotient spaces are linked 2-spheres.

Compact Kähler manifolds with positive curvature have contractible covers.

We define a new spectrum for compact length spaces and Riemannian manifolds called the "covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers $δ>0$ which identify the distinct $δ$ covers of the space. We investigat…

The paper proves conditions for self-covering manifolds to be fiber bundles over a circle.

Han discusses variants of digital covering maps and their equivalences.

New examples show transverse knots are determined by their branched covers.

Open manifolds can be covered by $\mathbb{R}^m$ with finite or infinite degree.

Computes constants for cyclic covers of translation surfaces.

The paper studies orbifold splice quotients and log covers of surface pairs.

To a branched cover between closed, connected and orientable surfaces one associates a "branch datum", which consists of the two surfaces, the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann-Hurwitz formula. A "candidate su…

Berestovskii and Plaut introduced the concept of a coverable uniform space when developing their theory of generalized universal covering maps for uniform spaces. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a locally uniformly joinable uniform space when developing their theory of generalized uniform co…

We prove the homotopy invariance of L^2 torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the L^2 cohomology of the covering space vanishes, the homotopy invariance was establi…

We provide criteria ensuring that a tunnel number one knot $K$ is not determined by its double branched cover, in the sense that the double branched cover is also the double branched cover of a knot $K'$ not equivalent to $K$.

The paper classifies coverings and non-Hausdorff extensions of Misner spacetime.

A branched covering surface-knot is a surface-knot in the form of a branched covering over an oriented surface-knot $F$, where we include the case when the covering has no branch points. A branched covering surface-knot is presented by a graph called a chart on a surface diagram of $F$. We can simplify a branched cover…