Гиперболические объемы конических многообразий для двумостовых узлов

We investigate the existence of hyperbolic, spherical or Euclidean structure on cone-manifolds whose underlying space is the three-dimensional sphere and singular set is a given two-bridge knot. For two-bridge knots with not more than 7 crossings we present trigonometrical identities involving the lengths of singular geodesics and cone angles of such cone-manifolds. Then these identities are used to produce exact integral formulae for the volume of the corresponding cone-manifold modeled in the hyperbolic, spherical and Euclidean geometries.

We give an overview of recent results on the geometry of knots and links. More precisely, we investigate the existence of hyperbolic, spherical or Euclidean structure on various cone manifolds whose underlying space is the three-dimensional sphere and whose singular set is a given knot or link. We present trigonometrical identities involving the lengths of singular geodesics and cone angles of such cone manifolds. Then these identities are used to produce exact integral formulas for volumes of the corresponding cone manifolds.

In this paper we show that all Lins-Mandel spaces S (b, l, t, c) are branched cyclic coverings of the 3-sphere. When the space is a 3-manifold, the branching set of the covering is a two-bridge knot or link of type (l, t) and otherwise is a graph with two vertices joined by three edges (a θ-graph). In the latter case the singular set of the space is always composed by two points with homeomorphic links. The first homology groups of the Lins-Mandel manifolds are computed when t=1 and when the branching set is a knot of genus one. Furthermore the family of spaces has been extended in order to contain all branched cyclic coverings of two-bridge knots or links.

For every group [math] , we define the set of hyperbolic structures on [math] , denoted by [math] , which consists of equivalence classes of (possibly infinite) generating sets of [math] such that the corresponding Cayley graph is hyperbolic; two generating sets of [math] are equivalent if the corresponding word metrics on [math] are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded [math] –actions on hyperbolic spaces. We are especially interested in the subset [math] of acylindrically hyperbolic structures on [math] , ie hyperbolic structures corresponding to acylindrical actions. Elements of [math] can be ordered in a natural way according to the amount of information they provide about the group [math] . The main goal of this paper is to initiate the study of the posets [math] and [math] for various groups [math] . We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of [math] , and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.

The rank one symmetric spaces of negative curvature come in three infinite families: real hyperbolic space H; complex hyperbolic space CH; and quaternionic hyperbolic space QH. (The Cayley plane is the remaining example.) Aside from the obvious embeddings H ↪→ CH ↪→ QH the three geometries seem fairly unrelated to each other. For instance, H admits non-arithmetic lattices in all dimensions [GrP] while QH only admits arithmetic lattices [GrS]. (See [C] for a related result.) The question of non-arithmetic lattices in CH is a basic unsolved problem [DM]. For a representation-theoretic comparison of discrete subgroups in the different rank one spaces, see [Sh]. In this paper we make a new connection between H and CH. We construct a closed hyperbolic 3-manifold which (as a diffeomorphic copy) is the ideal boundary of a complex hyperbolic 4-manifold. Up to index 2, the isometry group of CH is PU(2, 1), the group of complex projective automorphisms of the unit ball in C. The ideal boundary of CH is the unit 3-sphere S. A spherical CR structure on a 3-manifold is a system of coordinate charts into S whose transition functions are restrictions of elements of PU(2, 1). While plenty of closed Seifert fibered manifolds admit spherical CR structures [KT], our example gives the only known spherical CR structure on a closed hyperbolic 3-manifold.

In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3 -space H 3 . Galvez, Martinez and Milan showed that when the singular set does not accumulate at an end, the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called pitch p ) of the end determines the limiting shape, even when the singular set does accumulate at the end. If the singular set is bounded away from the end, we have - 1 < p ≤ 0 . If the singular set accumulates at the end, the pitch p is a positive rational number not equal to 1 . Choosing appropriate positive integers n and m so that p = n / m , suitable slices of the end by horospheres are asymptotic to d -coverings ( d -times wrapped coverings) of epicycloids or d -coverings of hypocycloids with 2 n 0 cusps and whose normal directions have winding number m 0 , where n = n 0 d , m = m 0 d ( n 0 , m 0 are integers or half-integers) and d is the greatest common divisor of m - n and m + n . Furthermore, it is known that the caustics of flat surfaces are also flat. So, as an application, we give a useful explicit formula for the pitch of ends of caustics of complete flat fronts.

The paper deals with the study of flat fronts in the hyperbolic 3-space, \(\mathbb {H}^3\). We characterize when an analytic curve of \(\mathbb {H}^3\) is in the singular set of some flat front with prescribed cuspidal edges and swallowtail singularities. We also prove that every complete flat front with a non-degenerate analytic planar singular set must be rotational.

Regenerating hyperbolic and spherical cone structures from Euclidean ones

The need to understand the structure of hierarchical or high-dimensional data is present in a variety of fields. Hyperbolic spaces have proven to be an important tool for embedding computations and analysis tasks as their non-linear nature lends itself well to tree or graph data. Subsequently, they have also been used in the visualization of high-dimensional data, where they exhibit increased embedding performance. However, none of the existing dimensionality reduction methods for embedding into hyperbolic spaces scale well with the size of the input data. That is because the embeddings are computed via iterative optimization schemes and the computation cost of every iteration is quadratic in the size of the input. Furthermore, due to the non-linear nature of hyperbolic spaces, euclidean acceleration structures cannot directly be translated to the hyperbolic setting. This article introduces the first acceleration structure for hyperbolic embeddings, building upon a polar quadtree. We compare our approach with existing methods and demonstrate that it computes embeddings of similar quality in significantly less time. Implementation and scripts for the experiments can be found at https://graphics.tudelft.nl/accelerating-hyperbolic-tsne.

In this paper we define a new invariant of the incomplete hyperbolic structures on a 1–cusped finite volume hyperbolic 3–manifold [math] , called the ortholength invariant. We show that away from a (possibly empty) subvariety of excluded values this invariant both locally parameterises equivalence classes of hyperbolic structures and is a complete invariant of the Dehn fillings of [math] which admit a hyperbolic structure. We also give an explicit formula for the ortholength invariant in terms of the traces of the holonomies of certain loops in [math] . Conjecturally this new invariant is intimately related to the boundary of the hyperbolic Dehn surgery space of [math] .

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space -- the natural extension of hyperbolic space by the de Sitter space -- except for the degenerate case where all simplices are Euclidean in a generalized sense. Those extended hyperbolic structures can realize geometrically a decomposition of the manifold as connected sum, along embedded spheres (or projective planes) which are totally geodesic, space-like surfaces in the de Sitter part of the extended hyperbolic structure.

The purpose of this note is to discuss examples of geometric transition from hyperbolic structures to half-pipe and Anti-de Sitter structures in dimensions two, three and four. As a warm-up, explicit examples of transition to Euclidean and spherical structures are presented. No new results appear here; nor an exhaustive treatment is aimed. On the other hand, details of some elementary computations are provided to explain certain techniques involved. This note, and in particular the last section, can also serve as an introduction to the ideas behind the four-dimensional construction of [RS19].

In the previous paper, Robert Riley [4] and his computer file Poincare found a fundamental domain for the action of a discrete group G of isometries of hyperbolic space HI generated by three parabolics. In this paper, we show that the orbit space H3/G is homeomorphic to a complement S3 k*, where k* is k union a point and where k is the (3, 3, 3) pretzel knot. Furthermore, H 3/G is equipped with an infinite volume hyperbolic orbifold structure. This should not be confused with the complete finite volume hyperbolic structure on S3 k. A neighborhood of k in H 3/G is not the quotient of a horoball by a group of parabolic isometries with common fixed point. It is instead the quotient of a neighbourhood of the domain of discontinuity for G. In addition, G has elliptic elements of order three which give rise to three singular axes in the hyperbolic structure. These three axes meet at a point at infinity in H 3/G. This accounts for the additional missing point in 3k*. A neighbourhood of this point is the quotient of a horoball by the Euclidean (3,3,3) triangle group. For a treatment of hyperbolic and orbifold structures, see Thurston [5]. The computer output form Riley's program consists of a picture of the fundamental domain 6D, and a data output giving the face pairings and face pairing transformations. For this paper, only the picture of 6D appearing in Section 3 of the preceding paper is used, as the information that it contains is sufficient to determine the face pairings uniquely. As a result, our labellings are different from those appearing in the data output, but the face pairings are the same. To show the homeomorphism H03/G = 3k*, we will glue up by identifying paired faces. The geometric structure will then arise as a direct consequence of the gluing. The domain 6X is an infinite volume hyperbolic polyhedron. In the upper half space model G&3 it lies between two EH-planes parallel to the imaginary axis of the boundary complex plane iro. The E-closure of 6D, GD, contains three subsets of i0. Two of these are compact, and are labelled Y and Z. The third region, labelled X, has connected closure in ro* and intersects any neighborhood of the point { oo Figure 1 shows a slightly altered version of the original computer drawing with added labellings of faces and edges. Some of the edges of 6D are EH-lines and, as such, they do not appear on the computer drawing. This poses no difficulty, as these edges will be subsumed by the first gluing step and will play no further part in the discussion.

In recent years, game developers are interested in developing games in the hyperbolic space. Shape blending is one of the fundamental techniques to produce animation and videos games. This paper presents two algorithms for blending between two closed curves in the hyperbolic plane in a manner that guarantees that the intermediate curves are closed. We deal with hyperbolic discrete curves on Poincaré disc which is a famous model of the hyperbolic plane. We use the linear interpolation approach of the geometric invariants of hyperbolic polygons namely hyperbolic side lengths, exterior angles and geodesic discrete curvature. We formulate the closing condition of a hyperbolic polygon in terms of its geodesic side lengths and exterior angles. This is to be able to generate closed intermediate curves. Finally, some experimental results are given to illustrate that the proposed methods generate aesthetic blending of closed hyperbolic curves.

We study cone-manifolds whose singular set is the trefoil knot with a bridge and whose underlying space is the 3-dimensional sphere. We also establish necessary and sufficient conditions for the existence of such manifolds in both Euclidean and hyperbolic geometries, and derive explicit volume formulas in each case.