arXiv research

Constructs a topological cover of real line's multiplicative group.

Novikov homology refined for 2D integer group coverings.

Study determines left-orderable properties of knot covers.

We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology $3$-spheres and cyclic branched covers of closed braids. The latter allows us to complete the proof…

The paper proves every closed curve on a bouquet of circles lifts to finite covers.

In a recent paper Y. Hu has given a sufficient condition for the fundamental group of the r-th cyclic branched covering of S^3 along a prime knot to be left-orderable in terms of representations of the knot group. Applying her criterion to a large class of two-bridge knots, we determine a range of the integer r>1 for w…

For each integer $d$ at least two, we construct non-spin closed oriented flat manifolds with holonomy group $\mathbb Z_2^d$ and with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have trivial Stiefel-Whitney classes.

Six quaternionic lines with optimal angles found in 2D quaternion space.

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 smoothly knotted n-sphere in the (n+2)-sphere for all n at le…

Study on left orderability of specific knot covers.

Arithmetic study of knots connects homology and SL2 representations.

Study proves non-left-orderability of 3-manifolds derived from specific knots.

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $d$ and $r$ such that $4\leq r \leq 2d^2-2d$, there is a non-singular hyperbolic curve of degree $2d$ in $\mathbb R^2$ with exactl…

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose maximal Euclidean factor has dimension $k$, then $π_1(M)$ is finitely generated. In p…

We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since d…

Study irreducible SU(2) representations for knots in 3D.

The paper proves that certain surface mapping class groups do not virtually surject to the integers.

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give sever…

We show that every strongly-cyclic branched covering of a (1,1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds coincides with the class of strongly-cyclic branched coverings of (1,1)-knots. As a c…

The paper studies automorphisms of Riemann surfaces and their covers.

We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g_1,...,g_k of length at most 2D and relators of the form g_ig_m = g_j . In particular, we obtain an explicit bound for the number k of generators in terms of the number "short loops" …

Study 2D viscoelastic equations using Lie group theory.

New findings show infinitely many non-homeomorphic manifolds with same proper homotopy type.

We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice $Γ$, applying a classical result of Macbeath to a suitable $Γ$-invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups ${\rm PU}(2,1,\mathcal{O…

Study on pretzel knots and their left-orderable properties.

Let $f:S^2\to S^2$ be an orientation-preserving branched covering map of degree $d\geq 2$, and let $Σ$ be an oriented Jordan curve passing through the critical values of $f$. Then $Γ:=f^{-1}(Σ)$ is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs a…

Updated rerefences and introduction. Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the Casson-Walker invariants of these 3-manifolds in terms …

We show that, for any prime p, a knot K in the 3-sphere is determined by its p-fold cyclic unbranched covering. We also investigate when the m-fold cyclic unbranched covering of a knot coincides with the n-fold cyclic unbranched covering of another knot, for different coprime integers m and n.

We exhibit new examples of double Kodaira fibrations by using finite Galois covers of a product $Σ_b \times Σ_b$, where $Σ_b$ is a smooth projective curve of genus $b \geq 2$. Each cover is obtained by providing an explicit group epimorphism from the pure braid group $\mathsf{P}_2(Σ_b)$ to some finite Heisenberg group.…

The existence or non-existence of Einstein metrics on 4-manifolds with non-trivial fundamental group and the relation with the underlying differential structure are analyzed. For most points $(n,m)$ in a large region of the integer lattice, the manifold $n\mathbb C \mathbb P^2\#m \overline{\mathbb C \mathbb P^2}$ is sh…

Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the sequence of torsion numbers satisfies a linear homogeneous recurrence relation …

We construct an infinite tower of covering spaces over the configuration space of $n-1$ distinct non-zero points in the complex plane. This results in an action of the braid group $\mathbb{B}_n$ on the set of $n$-adic integers $\mathbb{Z}_n$ for all natural numbers $n\geq 2$. We study some of the properties of these ac…

Researchers compute cohomology of mapping class groups with Prym representations, showing instability for large genus.

This paper is devoted to obtain the one-dimensional group invariant solutions of the two-dimensional Ricci flow ((2D) Rf) equation. By classifying the orbits of the adjoint representation of the symmetry group on its Lie algebra, the optimal system of one-dimensional subalgebras of the ((2D) Rf) equation is obtained. F…

Cyclic covers of knots uniquely determine the original knot.

Loi and Piergallini showed that a smooth compact, connected $4$-manifold $X$ with boundary admits a Stein structure if and only if $X$ is a simple branched cover of a $4$-disk $D^4$ branched along a positive braided surface $S$ in a bidisk $D_{1}^{2} \times D_{2}^{2} \approx D^4$. For each integer $N \geq 2$, we constr…

Study horocycle orbits in $\mathbb{Z}$-covers of hyperbolic surfaces.

The paper proves smoothness of stationary varifolds.

We show that if the branched double cover of an alternating link arises as $p/q \in \mathbb{Q} \setminus \mathbb{Z}$ surgery on a knot in $S^3$, then this is exhibited by a rational tangle replacement in an alternating diagram.

Let $x\in\mathbb{R}^{n}$. For $φ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $φ^{t}=t^{-1}φ(xt)$. A projective flow is a solution to the projective translation equation $φ^{t+s}=φ^{t}\circφ^{s}$, $t,s\in\mathbb{R}$. Previously we have developed an arithmetic, topologic and analytic theory of $2$-d…

The paper shows some Montesinos links can't be doubly sliced strongly.

We apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with half-densities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov-Kashiwara index. As in th…

We study the contribution of multiple covers of an irreducible rational curve C in a Calabi-Yau threefold Y to the genus 0 Gromov-Witten invariants in the following cases. (1) If the curve C has one node and satisfies a certain genericity condition, we prove that the contribution of multiple covers of degree d is given…

We apply the OSCAR (octagonal selection and clustering algorithms for regression) in recovering group-sparse matrices (two-dimensional---2D---arrays) from compressive measurements. We propose a 2D version of OSCAR (2OSCAR) consisting of the $\ell_1$ norm and the pair-wise $\ell_{\infty}$ norm, which is convex but non-d…

Affirmative answer to splitting question for open manifolds with nonnegative Ricci curvature and linear volume growth.

A preliminary group classification of the class 2D nonlinear heat equations $u_t=f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})$, where $f$ is arbitrary smooth function of the variables $x,y,u,u_x$ and $u_y$ using Lie method, is given. The paper is one of the few applications of an algebraic approach to the problem of group classific…

For a commutative ring $R$ with a unit, an $R$-homology rose is a topological space whose homology groups with $R$-coefficients agree with those of a bouquet of cirlces. In this paper, we study some special properties of covering spaces and fundamental groups of $R$-homology roses, from which we obtain some result supp…

For integers $d \geq 2$ and $ε= 0$ or 1, let $S^{1, d - 1}(ε)$ denote the sphere product $S^{1} \times S^{d - 1}$ if $ε= 0$ and the twisted $S^{d - 1}$ bundle over $S^{1}$ if $ε= 1$. The main results of this paper are: (a) if $d \equiv ε$ (mod 2) then $S^{1, d - 1}(ε)$ has a unique minimal triangulation using $2d + 3$ …