arXiv research

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

Cyclic covers of knots uniquely determine the original knot.

We study the linking numbers in a rational homology 3-sphere and in the infinite cyclic cover of the complement of a knot. They take values in $\Bbb Q$ and in ${Q}({\Bbb Z}[t,t^{-1}])$ respectively, where ${Q}({\Bbb Z}[t,t^{-1}])$ denotes the quotient field of ${\Bbb Z}[t,t^{-1}]$. It is known that the modulo-$\Bbb Z$ …

In this paper we show that periodic Takahashi 3-manifolds are cyclic coverings of the connected sum of two lens spaces (possibly cyclic coverings of the 3-sphere), branched over knots. When the base space is a 3-sphere, we prove that the associated branching set is a two-bridge knot of genus one, and we determine its t…

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

Links with isotopic preimages in $S^3$ are isotopic in $RP^3$.

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minim…

Study determines left-orderable properties of knot covers.

Study $p$-torsion growth in covers of 3-manifolds, proving Iwasawa formulas.

We prove a conjecture of Przytycki which asserts that the $n$-quandle of a link $L$ in the 3-sphere is finite if and only if the fundamental group of the $n$-fold cyclic branched cover of the 3-sphere, branched over $L$, is finite.

New graph invariant measures embeddability in 3D.

For any alternating knot, it is known that the double branched cover of the $3$-sphere branched over the knot is an $L$-space. We show that the three-fold cyclic branched cover is also an $L$-space for any genus one alternating knot.

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.

Proof of genus formula for 3-manifolds using arithmetic topology.

New invariant for links in 3-sphere computed and computed using diagrams.

Study symplectic cobordisms to empty sets, called hats, in 3-sphere and blow-ups.

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere …

Formula for Lefschetz number of knot branched covers.

The paper calculates properties of weaving knots and their unknotting numbers.

A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over the 3-sphere from some triangulation of S^3. This result is related to a theorem …

3-manifolds are chiral if not finitely covered by sphere or product.

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…

One method for obtaining every closed orientable 3-manifold is as branched covering of the 3-sphere over a link. There is a classical topological result showing that the minimun possible number of sheets in the covering is three. In this paper we obtain a geometric version of this result. The interest is given by the g…

We show that there exists a transverse link in the standard contact structures on the 3-sphere such that all contact 3-manifolds are contact branched covers over this transverse link.

It is known that if any prime power branched cyclic cover of a knot in the 3-sphere is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in the 3-sphere whose prime power branched cyclic covers are homology spheres. We show …

Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they …

Infinitely many hyperbolic links in lens space have isotopic lifts in 3-sphere.

Study finds symplectic fillings' properties for specific contact covers.

Recently, Mullins calculated the Casson-Walker invariant of the 2-fold cyclic branched cover of an oriented link in S^3 in terms of its Jones polynomial and its signature, under the assumption that the 2-fold branched cover is a rational homology 3-sphere. Using elementary principles, we provide a similar calculation f…

We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic…

By a construction of Berstein and Edmonds every proper branched cover f between manifolds is a factor of a branched covering orbit map from a locally connected and locally compact Hausdorff space called the monodromy space of f to the target manifold. For proper branched covers between 2-manifolds the monodromy space i…

Study four ways to define braided open book decompositions of the 3-sphere.

The study of symmetries in manifolds derived from colored polytopes.

In this paper we study equivariant constrained Willmore tori in the 3-sphere. These tori admit a 1-parameter group of Möbius symmetries and are critical points of the Willmore energy under conformal variations. We show that the associated spectral curve of an equivariant torus is given by a double covering of $\mathbb …

We show that the fundamental group of the double branched cover of an infinite family of homologically thin, non-quasi-alternating knots is not left-orderable, giving further support for a conjecture of Boyer, Gordon, and Watson that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental…

We show that every p-fold strictly-cyclic branched covering of a b-bridge link in the 3-sphere admits a p-symmetric Heegaard splitting of genus g=(b-1)(p-1). This gives a complete converse to a result of Birman and Hilden, and gives an intrinsic characterization of p-symmetric Heegaard splittings as p-fold strictly-cyc…

We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions. Using these descriptions, we give presentations for their fundamental groups, w…

We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3-sphere, branched over two-bridge knots. Our method is to use the bi-twisted face-pairing constructions of Cannon, Floyd, and Parry; these examples show that the bi-twist construction is often efficient and natural…

Given a 3-manifold $Y$ and a free homotopy class in $[S^1,Y]$, we investigate the set of topological concordance classes of knots in $Y \times [0,1]$ representing the given homotopy class. The concordance group of knots in the 3-sphere acts on this set. We show in many cases that the action is not transitive, using two…

We prove a folklore theorem of W. Thurston which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot U is prime if and only if lifting the third arc of the theta-curve to the double branched cover over U …

We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These examples come from a tower of abelian covers of an explicit arithmetic 3-manifold.…

In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the $3$-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid con…

We consider closed orientable 3-dimensional hyperbolic manifolds which are cyclic branched coverings of the 3-sphere, with branching set being a two-bridge knot (or link). We establish two-sided linear bounds depending on the order of the covering for the Matveev complexity of the covering manifold. The lower estimate …

Based on the analogy between knots and primes, J. Hillman, D. Matei and M. Morishita defined the Iwasawa invariants for sequences of cyclic covers of links with an analogue of Iwasawa's class number formula of number fields. In this paper, we consider the existence of covers of links with prescribed Iwasawa invariants,…

Let $p$ be a prime number. We develop a theory of $p$-adic Mahler measure of polynomials and apply it to the study of $\mathbb{Z}$-covers of rational homology 3-spheres branched over links. We obtain a $p$-adic analogue of the asymptotic formula of the torsion homology growth and a balance formula among the leading coe…

Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical proje…

An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link $L$ in $L(p,q)$, that is the counterimage $\widetilde L$ of $L$ under the universal covering of $L(p,q)$. If lens spaces are defined as a lens with suitable boundary identifications, then a link in $L(p,q)$ can be represented…

We are interested in finite groups acting orientation-preservingly on 3-manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic …