arXiv research

Given a closed oriented PL four-manifold $X$ and a closed surface $B$ embedded in $X$ with isolated cone singularities, we give a formula for the signature of an irregular dihedral cover of $X$ branched along $B$. For $X$ simply-connected, we deduce a necessary condition on the intersection form of a simply-connected i…

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

Algorithm calculates ribbon obstructions for colored knots.

Consider a dihedral cover $f: Y\to X$ with $X$ and $Y$ four-manifolds and $f$ branched along an oriented surface embedded in $X$ with isolated cone singularities. We prove that only a slice knot can arise as the unique singularity on an irregular dihedral cover $f: Y\to S^4$ if $Y$ is homotopy equivalent to $\mathbb{CP…

Dihedral linking invariant uses knot colorings to distinguish knots.

The dihedral genus of a knot is related to its signature and minimal surface genus.

We consider knots equipped with a representation of their knot groups onto a dihedral group D_{2n} (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relati…

Sharp obstruction found for extending knot group quotients over surfaces in $B^4$.

Algorithms compute invariants of 4-manifolds as branched covers.

Let K be a knot in an integral homology 3-sphere and let B denote the 2-fold branched cover of the integral homology sphere branched along K. We construct a map from the slice of characters with trace free along meridians in the SL(2, C)-character variety of the knot exterior to the SL(2, C)-character variety of 2-fold…

We exhibit the traceless $SU(2)$ character variety of a 6-punctured 2-sphere as a 2-fold branched cover of ${\mathbb{C}}P^3$, branched over the singular Kummer surface, with the branch locus in $R(S^2,6)$ corresponding to the binary dihedral representations. This follows from an analysis of the map induced on $SU(2)$ c…

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…

Let $M$ be a connected, closed, oriented three-manifold and $K$, $L$ two rationally null-homologous oriented simple closed curves in $M$. We give an explicit algorithm for computing the linking number between $K$ and $L$ in terms of a presentation of $M$ as an irregular dihedral $3$-fold cover of $S^3$ branched along a…

New formula for signature of topological branched covers, extending classical results.

In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.

We show in this paper that the set of irreducible components of the family of Galois coverings of P^1_C with Galois group isomorphic to D_n is in bijection with the set of possible numerical types. In this special case the numerical type is the equivalence class (for automorphisms of D_n) of the function which to each …

The paper explores representations of specific knot groups and their properties.

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

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

Course on knots using branched coverings.

Formula compares metrics on branched coverings of line bundles.

New criterion for branched covers between 2-spheres.

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.

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…

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.

Branched covers between Riemann surfaces are associated with certain combinatorial data, and Hurwitz existence problem asks whether given data satisfying those combinatorial constraints can be realized by some branched cover. We connect recent development in spherical conic metrics to this old problem, and give a new m…

New CMC surfaces with dihedral symmetry constructed from Darboux transforms.

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

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.

Study geometric properties of branched covers of hyperbolic manifolds.

Cyclic covers of knots uniquely determine the original knot.

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

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…

A branched covering surface-knot over an oriented surface-knot $F$ is a surface-knot in the form of a branched covering over $F$. A branched covering surface-knot over $F$ is presented by a graph called a chart on a surface diagram of $F$. For a branched covering surface-knot, an addition of 1-handles equipped with cha…

Gromov-Thurston covers have Betti numbers as expected.

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

Many three dimensional manifolds are two-fold branched covers of the three dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and many examples. It shows that there are three dimensional homology spheres that do not two-fold branched cover any manifol…

Homotopy proof for pseudomanifolds via branched covers.

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…

Bianchi proves Mumford conjecture using branched covers.

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