Related papers: Linking topological spheres
We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with rational angles in degree: they are a one-parameter family of symmetric $a^4b$-pentagonal subdivisions of the tetrahedron with…
We show that the following algorithmic problem is decidable: given a $2$-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in $\mathbf{R}^3$? By a known reduction, it suffices to decide…
The classical Seifert algorithm provides an explicit construction of a Seifert surface for any link in $S^3$. Alegria and Menasco extended this construction to integral homology $3$-spheres using Heegaard splittings. In this paper, we…
We show that there exists a link with 2 components which is not smoothly slice in $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$. By contrast, it is well-known that every knot (i.e., link with 1 component) is smoothly slice therein. Our proof…
We construct a number of topologically trivial but smoothly non-trivial families of embeddings of 3-manifolds in 4-manifolds. These include embeddings of homology spheres in $S^4$ that are not isotopic but have diffeomorphic complements,…
A graph embedded in the 3-sphere is called irreducible if it is non-splittable and for any 2-sphere embedded in the 3-sphere that intersects the graph at one point the graph is contained in one of the 3-balls bounded by the 2-sphere. We…
Special class of surfaces in five-dimensional sphere in $C^3$ is considered. Immersion equations for minimal tori of that class are shown to be reducible to the equation $u_{z\bar z}=e^u-e^{-2u}$ which is integrable by means of inverse…
Let L be a link in the 3-sphere that is in thin position but not in bridge position and let P be a thin level sphere. We generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks that P can have,…
Let S be a finite union of (pairwise disjoint but possibly knotted and linked) closed curves and tubes in the round sphere S^3 or in the flat torus T^3. In the case of the torus, S is further assumed to be contained in a contractible subset…
A theory of signatures for odd-dimensional links in rational homology spheres is studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant under appropriately generalized concordance and a special…
We construct a contactomorphism of the standard sphere which does not have any translated points, providing a negative answer to a conjecture posed by Sandon.
A one parameter set of noncommutative complex algebras is given. These may be considered deformation quantisation algebras. The commutative limit of these algebras correspond to the algebra of polynomial functions over a manifold or…
We introduce the notion of locally consistent system of half-spaces for a real hyperplane arrangement. We embed a sphere in the complexified complement by shifting the real unit sphere into the imaginary direction indicated by the…
This article is covered by the article arxiv.1012.0925 We study intersection of two polyhedral spheres without self-intersections in 3-space. We find necessary and sufficient conditions on sequences x = x_1,x_2,...,x_n, y = y_1,y_2,...,y_n…
Consider the network of elements connected by adjacency in atomic number and periodic table column. If all elements except H-He are connected by identical springs and constrained to a z-spiral in 2D, a map sans spring-crossing results for…
We investigate certain $4$-dimensional analogues of the classical $3$-dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential…
Two spheres with centers $p$ and $q$ and signed radii $r$ and $s$ are said to be in contact if $|p-q|^2 = (r-s)^2$. Using Lie's line-sphere correspondence, we show that if $F$ is a field in which $-1$ is not a square, then there is an…
Given a sequence of oriented links L^1,L^2,L^3,... each of which has a distinguished, unknotted component, there is a decomposition of the 3-sphere naturally associated to it, which is constructed as the components of the intersection of an…
We prove that if a simplicial complex is shellable, then the intersection lattice for the corresponding diagonal arrangement is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on…
We show that the $\mathbb{Q}/\mathbb{Z}$-valued linking forms on rational homology spheres are (anti-) symmetric and we compute the linking form of a 3-dimensional rational homology sphere in terms of a Heegaard splitting. Both results have…