A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Characterizes billiard and quasigeodesic flows in polyhedral convex bodies.
New Jordan Floer homology shows every curve inscribes every isosceles trapezoid.
Continuous curves inscribe isosceles trapezoids in complex plane.
Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.
An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. We prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right, or 120-isosceles triangle; a 120-rhombus; a 60-90-120 kite; or a regular hexagon…
Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.
We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this …
We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra t…
We review the theory of intrinsic geometry of convex surfaces in the Euclidean space and prove the following theorem: if the surface of a convex body K contains arbitrary long closed simple geodesics, then K is an isosceles tetrahedron.
Survey on geodesics on tetrahedra in curved spaces.
Geodesic loops on tetrahedra are studied in spherical and hyperbolic spaces.
The paper connects quantum -symbols to tetrahedra volumes via discrete Fourier transforms.
The paper proves a Minkowski-like theorem for tetrahedra in dS3 and AdS3.
Randomly glued tetrahedra form connected 3-manifolds with a single boundary.
The present paper follows the computational approach to 3-manifold classification via edge-coloured graphs, already performed by several authors with respect to orientable 3-manifolds up to 28 coloured tetrahedra, non-orientable 3-manifolds up to 26 coloured tetrahedra, genus two 3-manifolds up to 34 coloured tetrahedr…
Motivated by the Turaev-Viro invariant of 3-manifolds, we construct a formal topological invariant of closed, oriented 3-manifolds involving spherical tetrahedra as an application of the asymptotic formula of 6j symbols for the Quantum Enveloping Algebra of sl(2). This invariant can be considered as a spherical version…
We classify the orientable finite-volume hyperbolic 3-manifolds having non-empty compact totally geodesic boundary and admitting an ideal triangulation with at most four tetrahedra. We also compute the volume of all such manifolds, we describe their canonical Kojima decomposition, and we discuss manifolds having cusps.…
Researchers found the minimum number of tetrahedra needed to triangulate elliptic and sol 3-manifolds.
Researchers found the Wigner derivative and its inverse are equal for spherical tetrahedra.
In the paper we define a "volume" for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedra complexes. We describe when this volume belongs to the Bloch group. In doing so, we recover and generalize results of Neumann-Zagier, N…
Hyperideal tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional topology, like the computation of quantum invariants of 3-manifolds and the use of …
We give a rigorous geometric proof of the Murakami-Yano formula for the volume of a hyperbolic tetrahedron. In doing so, we are led to consider generalized hyperbolic tetrahedra, which are allowed to be non-convex, and have vertices `beyond infinity'; and we uncover a group, which we call 22.5K, of 23040 scissors-class…
Hyperbolic links in thickened torus decompose into angled tetrahedra.
We analyze the dynamical properties of a tetrahedron transformation on the space of non-degenerate tetrahedra which can be identified with the non-compact globally symmetric -dimensional space $\mbox{Sl}(3,\mathbb{R}) / \mbox{So}(3,\mathbb{R})$. We establish the existence of a local attractor which coincides with th…
We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7,1) and L(7,2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e.algebraic relation corresponding to a ``2 tetrahedra to 3 t…
Census of 10-tetrahedra hyperbolic 3-manifolds with 150,730 new examples.
We improve and extend to the non-orientable case a recent result of Karabas, Malicki and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.
Study laws of cosines and sines for hyperbolic shapes with ideal vertices.
We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than and found the asymptotic of this number as goes to infinity.
In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. I…
A generalized hyperbolic tetrahedra is a polyhedron (possibly non-compact) with finite volume in hyperbolic space, obtained from a tetrahedron by the polar truncation at the vertices lying outside the space. In this paper it is proved that a volume formula for ordinary hyperbolic tetrahedra devised by J. Murakami and M…
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
The paper studies the face angles of tetrahedra with a fixed base.
Study - symbols linking anti-de Sitter tetrahedra to hyperbolic geometry.
We present a generalization of Minkowski's classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron's faces. This allows the reconstruction of both s…
Starting with an ideal triangulation of the interior of a compact 3-manifold M with boundary, no component of which is a 2-sphere, we provide a construction, called an inflation of the ideal triangulation, to obtain a strongly related triangulations of M itself. Besides a step-by-step algorithm for such a construction,…
In this paper we provide the first examples of non-flat soap films proven to span tetrahedra. These are members of a continuous two parameter family of soap films with tetrahedral boundaries. Of particular interest is a two parameter subfamily where each spanning soap film has the property that two minimal surfaces mee…
From its creation in 1989 through subsequent extensions, the widely-used "SnapPea census" now aims to represent all cusped finite-volume hyperbolic 3-manifolds that can be obtained from <= 8 ideal tetrahedra. Its construction, however, has relied on inexact computations and some unproven (though reasonable) assumptions…
Simple geodesics on spherical tetrahedra identified for specific angles.
Given a combinatorial description of a polyhedron having edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize is generally not a convex subset of \cite{DIAZ}. If has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angle…
Study on length spectrum of random hyperbolic 3-manifolds.
We prove that the number of combinatorially distinct causal 3-dimensional triangulations homeomorphic to the 3-dimensional sphere is bounded by an exponential function of the number of tetrahedra. It is also proven that the number of combinatorially distinct causal 4-dimensional triangulations homeomorphic to the 4-sph…
New examples show flip distance and polyhedron triangulation numbers differ, with ratio close to 3/2.
We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial tetrahedral tessellations with at most 25 (orientable case) and 21 (non-orienta…
We address the question of determining the eigenvalues (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), …
Several identities similar to the Schlaefli formula are established for tetrahedra in a space of constant curvature.
In this paper we are interested in computing representations of the fundamental group of a 3-manifold into PSL(3;C) (in particular in PSL(2;C); PSL(3;R) and PU(2; 1)). The representations are obtained by gluing decorated tetrahedra of flags. We list complete computations (giving 0-dimensional or 1-dimensional solution …
Consider a 3dimensional manifold obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a space of complete hyperbolic metrics on with cone singularities …