arXiv research

We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of the covering spaces to that of the base space. Finally, we show that this places restrictions on th…

This work identifies a class of moves on knots which translate to $m$-equivalences of the associated $p$-fold branched cyclic covers, for a fixed $m$ and any $p$ (with respect to the Goussarov-Habiro filtration.) These moves are applied to give a flexible (if specialised) construction of knots for which the Casson-Walk…

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 …

3-manifolds study Hasse norm principle, akin to number fields.

We show that a hyperbolic $3$-manifold can be the cyclic branched cover of at most fifteen knots in $\mathbf{S}^3$. This is a consequence of a general result about finite groups of orientation preserving diffeomorphisms acting on $3$-manifolds. A similar, although weaker, result holds for arbitrary irreducible $3$-mani…

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…

The paper studies liftable mapping class groups of cyclic covers of spheres.

We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generalization of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere.

We provide an alternative proof of a sufficient condition for the fundamental group of the $n^{th}$ cyclic branched cover of $S^3$ along a prime knot $K$ to be left-orderable, which is originally due to Boyer-Gordon-Watson. As an application of this sufficient condition, we show that for any $(p,q)$ two-bridge knot, wi…

The paper explores circular orderability in 3-manifold groups, related to the L-space conjecture.

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-$\Bbb Z$ …

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…

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 …

The paper studies mapping class groups of cyclic covers and their liftable counterparts.

Study of infinitesimal rigidity in hyperbolic manifolds.

New Alexander invariant classes computed for knot group representations.

We give a complete proof of Thurston's Orbifold Theorem for very good 3-orbifolds of cyclic type. An orbifold is said to be very good when it has a finite cover which is a manifold. A 3-orbifold is of cyclic type if the singular set is a non-empty 1-manifold transverse to the boundary.

Toroidal 3-manifolds have special group structures that can be shown through specific covers.

The Weil conjecture is a delightful theorem for algebraic varieties on finite fields and an important model for dynamical zeta functions. In this paper, we prove a functional equation of Lefschetz zeta functions for infinite cyclic coverings which is analogous to the Weil conjecture. Applying this functional equation t…

The authors conjectured previously that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from symbolic dynamics.

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,\…

New quantum invariant is asymptotically multiplicative under cyclic covers.

Cyclic covers of knots uniquely determine the original knot.

In this paper we study the connections between cyclic presentations of groups and branched cyclic coverings of (1,1)-knots. In particular, we prove that every n-fold strongly-cyclic branched covering of a (1,1)-knot admits a cyclic presentation for the fundamental group encoded by a Heegaard diagram of genus n.

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 examples show transverse knots are determined by their branched covers.

We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1,1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold strongly-cyclic branched coverings of (1,1)-knots, thro…

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundament…

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.

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…

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…

The paper studies cyclic covers of rational surfaces and their Hodge structures.

Study on pretzel knots showing cyclic branched covers are L-spaces.

Study disproves conjecture about quadratic differentials.

In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, $P^2$-irreducible 3-manifold corresponding to a non-trivial, finitely-generated subgroup of its fundamental group is infinite, then either the covering space is almost compact or the subgroup is infinite…

Computes constants for cyclic covers of translation surfaces.

The study proves residual finiteness for certain lattice extensions and negatively curved projective varieties.

Leighton's graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton's theorem that allows generalizations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups w…

Given a TQFT in dimension d+1, and an infinite cyclic covering of a closed (d+1)-dimensional manifold M, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams' work in symbolic dynamics. The Turaev-Viro module associated to…

Strongly-cyclic branched coverings of knots are studied by using their (g,1)-decompositions. Necessary and sufficient conditions for the existence and uniqueness of such coverings are obtained. It is also shown that their fundamental groups admit geometric g-words cyclic presentations.

In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\bf S^3$), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of …

We show that for any po sitive integer $m$, there exist order $n$ Stein corks. The boundaries are cyclic branched covers of slice knots embedded in the boundary of corks. By applying these corks to generalized forms, we give a method producing examples of many finite order corks, which are possibly not Stein cork.

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theor…

We give a formula of 3-dimensional invariant for a cyclic contact branched covering of the standard contact S^{3}.

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…

We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental group, and are not L-spaces.

Study on left orderability of specific knot covers.

We characterize the cyclic branched covers of the 2-sphere where every homeomorphism of the sphere lifts to a homeomorphism of the covering surface. This answers a question that appeared in an early version of the erratum of Birman and Hilden [2].