Related papers: Three-Dimensional Small Covers and Links
We define a Floer-homology invariant for links in $S^3$, and study its properties.
A combinatorial presentation of closed orientable 3-manifolds as bi-tricolored links is given together with two versions of a calculus via moves to manipulate bi-tricolored links without changing the represented manifold. That is, we…
This note is a sequel to our earlier paper of the same title [dg-ga/9710001] and describes invariants of rational homology 3-spheres associated to acyclic orthogonal local systems. Our work is in the spirit of the Axelrod-Singer papers,…
For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the…
We study symplectic structures on four-dimensional small covers. Our main result shows that every symplectic four-dimensional small cover is aspherical. We then classify symplectic small covers over products of two polygons, proving that…
Let $p$ be a prime number and let $d\in \mathbb{Z}_{>0}$. In this paper, following the analogy between knots and primes, we study the $p$-torsion growth in a compatible system of $(\mathbb{Z}/p^n\mathbb{Z})^d$-covers of 3-manifolds and…
We prove that contact homeomorphisms preserve characteristic foliations on surfaces in contact $3$-manifolds. More precisely, since the characteristic foliation is a singular $1$-dimensional foliation, we show that singular points are…
We study the existence of branched coverings between closed $3$-manifolds, with emphasis on universal knots and links. We prove that the only closed $3$-manifolds that admit a universal link are spherical. Furthermore, we distinguish…
The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This…
The author studied in \cite{shi} the covering map property of the local homeomorphism associating to the space of stability conditions over the $n$-Kronecker quiver. In this paper, we discuss the covering map property for stability…
An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the…
Some ball-quotient orbifolds are related by covering maps. We exploit these coverings to find infinite towers of orbifolds uniformized by the complex 2-ball and some orbifolds over K3 surfaces uniformized by the 2-ball. Corresponding…
We obtain a topological and equivariant classification of closed, connected three-dimensional Alexandrov spaces admitting a local isometric circle action. We show, in particular, that such spaces are homeomorphic to connected sums of some…
In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside $\mathbb{R} \times \mathbb{R}^3 \equiv \mathbb{R}^4$, which forms a triple. We want to define an ambient isotopic…
A 3D rep-tile is a compact 3-manifold $X$ in $\mathbb{R}^3$ that can be decomposed into finitely many pieces, each of which are similar to $X$, and all of which are congruent to each other. In this paper we classify all 3D rep-tiles up to…
We prove that Khovanov homology is an invariant of links in unparametrized $\mathbb{RP}^3$'s, i.e., oriented $3$-manifolds diffeomorphic to $\mathbb{RP}^3$. Along the way, we establish the functoriality of Khovanov homology for link…
We consider causal 3-dimensional triangulations with the topology of $S^2\times [0,1]$ or $D^2\times [0,1]$ where $S^2$ and $D^2$ are the two-dimensional sphere and disc, respectively. These triangulations consist of slices and we show that…
We derive an integral formula for the linking number of two submanifolds of the n-sphere S^n, of the product S^n x R^m, and of other manifolds which appear as "nice" hypersurfaces in Euclidean space. The formulas are geometrically…
To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to…
Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be the d-component link in a homology 3-sphere that results from performing 1/q-surgery on the last component. Results about the Alexander polynomial and twisted…