arXiv research

Authors prove a contact structure result using branched covers and overtwisted disks.

We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.

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

New transverse links solve contact manifold problems.

Study finds symplectic fillings' properties for specific contact covers.

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…

Study fractional twist behavior in branched covers, with applications to 3-manifolds.

A transverse knot exists in S^3 that covers all contact 3-manifolds.

Given a transverse link in the standard contact 3-sphere, we study the contact manifold that arises as a branched double cover of the sphere. We give a contact surgery description of such manifolds, which allows to determine the Heegaard Floer contact invariants for some of them. By example of the knots of Birman--Mena…

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…

Using the relation between Khovanov homology and the Heegaard Floer homology of branched double covers, we show how Khovanov homology can be used to establish tightness of branched double covers of certain transverse knots. We give examples of several infinite families of knots whose branched covers are tight for Khova…

Study knot contact homology and character varieties to prove a conjecture about knots.

New examples show high twisting doesn't guarantee open book maximality.

We consider the problem of realizing tight contact structures on closed orientable three-manifolds. By applying the theorems of Hofer et al., one may deduce tightness from dynamical properties of (Reeb) flows transverse to the contact structure. We detail how two classical constructions, Dehn surgery and branched cover…

Torus knots $(4,5)$ and $(5,6)$ provide counterexamples to a conjecture about knot contact homology.

This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u_+ and u_- in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit gamma, the total multiplicity of the negative ends of u_+ at covers of gamma agrees with the total…

New knot invariant from equivariant Heegaard Floer cohomology.

Study classifies twist knots with maximal self-linking number in S^3.

We define the reduced Khovanov homology of an open book (S,h), and we identify a distinguished "contact element" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,h). Our construction generalizes the relationship between the reduced Khovanov homology …

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary ob…

We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We calculate the knot invariant for two-bridge knots and relate it to double branc…

For surface-knots, branched covers with degree 3 have simplifying numbers <3.

Criteria found for knots not determined by their double branched covers.

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

We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover…

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

Suppose S is a compact surface with boundary, and let g be a diffeomorphism of S which fixes the boundary pointwise. We denote by (M_{S,g},ξ_{S,g})$ the contact 3-manifold compatible with the open book (S,g). In this article, we construct a Stein cobordism from the contact connected sum (M_{S,h},ξ_{S,h}) # (M_{S,g},ξ_{…

Course on knots using branched coverings.

New method finds unrealizable branched cover data.

Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we generalize the obstruction bundle technique of Taubes for determining multiple cove…

Formula compares metrics on branched coverings of line bundles.

Simplifies surface-knots using chart moves involving black vertices.

New criterion for branched covers between 2-spheres.

The study shows knots from 3-braids cannot be concordant to a specific Legendrian unknot.

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

Given a branched covering of degree d between closed surfaces, it determines a collection of partitions of d, the branch data. In this work we show that any branch data are realized by an indecomposable primitive branched covering on a connected close surface N with Euler's characteristic less than or equal to 0. This …

Proper branched coverings are homeomorphisms on 3D balls or when branch set is empty.

In this work we characterize branch data of branched coverings of even degree over the projective plane which are realizable by indecomposable branched coverings.

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…

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

Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.

Taut foliations map leaves to branched 2-sphere covers.

Open and discrete maps with specific branch set images are equivalent to PL branched covers.

Simplifies surface-knots by adding 1-handles with chart loops.

Study branched coverings of singular (G,X)-manifolds, solving open questions.

Researchers study simple branched coverings and their cobordism groups.

Study geometric properties of branched covers of hyperbolic manifolds.

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.