Long geodesics imply a special shape of convex bodies.
problem Understanding the geometry of convex surfaces.
method Intrinsic geometry of convex surfaces and proof by contradiction.
result Long geodesics on a convex surface imply the shape is an isosceles tetrahedron.
New Jordan Floer homology shows every curve inscribes every isosceles trapezoid.
problem Understanding inscriptions of isosceles trapezoids in Jordan curves.
method Constructing a new Lagrangian Floer homology chain complex.
result Establishes new cases of non-smooth Jordan curves inscribing isosceles trapezoids.
Continuous curves inscribe isosceles trapezoids in complex plane.
problem Proving periodic curves inscribe isosceles trapezoids.
method Lagrangian intersection problem and convergence argument.
result Continuous curves inscribe trapezoids with any similarity type.
Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.
problem Finding the minimum of the spectral determinant on isosceles triangles.
method Analyzing the determinant of the Laplacian on Euclidean isosceles triangle envelopes of fixed area.
result Equilateral triangle envelope minimizes the determinant of the Laplacian.
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…
Characterizes billiard and quasigeodesic flows in polyhedral convex bodies.
problem Characterizing billiard and quasigeodesic flows in polyhedral convex bodies.
method Alexandrov geometry methods.
result Optimal regularity result for convex bodies: billiard dynamics is continuous if boundary is of class C2,1. First working model of a monostable tetrahedron described.
problem None explicitly stated in the abstract.
method Described a new model of a monostable tetrahedron.
result First working model of a monostable tetrahedron.
Proves volume conjecture for double twist knots using complexified tetrahedrons.
problem Volume conjecture for double twist knots
method Complexified tetrahedron and associated SL(2, C) representation of fundamental group
result Volume conjecture proved for double twist knots
Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.
problem Understanding periodic billiard paths in right triangles and orbit closures in strata of Abelian and quadratic differentials.
method Classifying orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials.
result Computed orbit closures and deduced asymptotic number of periodic billiard trajectories in right triangles.
New Bailey pairs derived for tetrahedron index, linking knot invariants.
problem Deriving knot invariants using Bailey pairs for the tetrahedron index.
method Developed new Bailey pairs in terms of q-series.
result New Bailey pairs express the pentagon identity of the tetrahedron index.
Paper shows a surface with positive genus under a tetrahedron, less than cone's area.
problem Existence of area-minimizing surfaces with positive genus spanning a tetrahedron.
method Triple cover approach, using BV functions and Caccioppoli sets.
result Found a surface of positive genus with area less than a conic surface.
Quantum theory of curved tetrahedrons yields quantum group intertwiners.
problem Quantum geometry of curved tetrahedrons and their intertwiners.
method Combinatorial quantization of tetrahedron phase space, relating to SU(2) flat connections.
result Physical Hilbert space coincides with Uq(su(2)) intertwiners, consistent with LQG area spectrum.
The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation through angle π in the middle points of a certain pair of its skew edges.
Quantum 6j-symbols linked to tetrahedra angles and volumes.
problem Understanding quantum 6j-symbols and their geometric interpretation. method Establishing the geometric connection between quantum 6j-symbols and tetrahedra angles, including spherical, Euclidean, and hyperbolic cases. result Quantum 6j-symbols correspond to dihedral angles of specific tetrahedra, including generalized hyperbolic ones, with exponential growth rates tied to their volumes. The study finds a local attractor for tetrahedron transformations.
problem Analyzing the dynamical properties of tetrahedron transformations.
method Analysis of a geometric tetrahedron transformation on a specific space.
result Existence of a local attractor coinciding with the set of regular tetrahedra.
Knots can be embedded into fractals like the Menger Sponge and Sierpinski Tetrahedron.
problem Embedding knots into fractals to compare complexity.
method Proved all knots can be embedded into Menger Sponge, and Pretzel knots into Sierpinski Tetrahedron. Compared iterations needed for given knots.
result Comparison of fractal complexity through knot embedding.
Complex b-6j symbols relate to hyperbolic tetrahedron volumes and determinants.
problem Analyzing asymptotics of complex b-6j symbols. method Relating asymptotics to hyperbolic tetrahedron volumes and determinants.
result Complex b-6j symbols' asymptotics linked to tetrahedron volumes and determinants. The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…
Simple geodesics on spherical tetrahedra identified for specific angles.
problem Identifying simple closed geodesics on regular tetrahedra in spherical space.
method Analyzing pairs of coprime integers (p,q) to find angles α1 and α2.
result Existence and non-existence of simple closed geodesics for specific angles.
Develops quantum cluster algebra approach to solve tetrahedron equation.
problem Investigates a three-dimensional generalization of the Yang-Baxter equation.
method Quantum cluster algebra approach with realization of quantum Y-variables in terms of q-Weyl algebras.
result Obtains a solution with three spectral parameters and reproduces Sergeev's R matrix.
We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron. Two cases are considered: a mildly truncated and a prism truncated tetrahedron. The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the correspondi…
Machine learning finds Z/2 eigenfunctions on a sphere.
problem Finding Z/2 eigenfunctions on the sphere.
method Created a multivalued neural network and used JAX to implement it. Fixed branch points at tetrahedron and cube vertices, and allowed AI to move them in the third case.
result Found Z/2 eigenfunctions for three cases.
The present paper regards the volume function of a doubly truncated hyperbolic tetrahedron. Starting from the previous results of J. Murakami, U. Yano and A. Ushijima, we have developed a unified approach to express the volume in different geometric cases via dilogarithm functions and to treat properly the many analyti…
The set of maximal non-integrable structures (SU(2)×SU(2),B,I), where B is Killing-Cartan metric is described as subset of CP3. The visualization of complex projective space CP3 as tetrahedron which edges and faces are CP1 and CP2 is used.
Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.
problem Analyzing projective structures on hyperbolic 3-orbifolds.
method Parameterization using classical invariants, traces, and geometric cross ratios.
result Computed and analyzed the moduli space of projective structures.
Models for 3D harmonic 1-forms and spinors near singular points.
problem Constructing models for Z/2 harmonic 1-forms and spinors in 3D near singular points. method Using symmetries of tetrahedron, octahedron, and icosahedron to construct local models on R3. result Local models are Z/2 harmonic 1-forms or spinors on R3 with zero locus consisting of rays from the origin. Through computer enumeration with the aid of topological results, we catalogue all 18 closed non-orientable P^2-irreducible 3-manifolds that can be formed from at most eight tetrahedra. In addition we give an overview as to how the 100 resulting minimal triangulations are constructed. Observations and conjectures are d…
New solutions to 3D integrability equations using quantum cluster algebras.
problem Constructing solutions to the tetrahedron and 3D reflection equations.
method Extending quantum cluster algebra approach to Fock-Goncharov quivers and investigating cluster transformations.
result Explicit formulas for matrix elements of solutions derived for typical representations.
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…
Investigates quantum 6j symbols for hyperbolic tetrahedra.
problem Determining asymptotics of quantum 6j symbols for specific tetrahedra. method Analyzes quantum 6j symbols for hyperbolic tetrahedra with ideal or ultra-ideal vertices, calculating the first two leading terms. result First two leading terms of quantum 6j symbols are given by volume and determinant of Gram matrix. 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…
This study finds the shape of centrally symmetric octahedra with specific angles.
problem Finding the shape of centrally symmetric octahedra with prescribed cone-deficits.
method Inspired by Thurston's work, the paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits, showing it forms a real hyperbolic ideal tetrahedron.
result The set of shapes of centrally symmetric octahedra with prescribed cone-deficits forms a real hyperbolic ideal tetrahedron with dihedral angles half of the prescribed cone-deficits.
Researchers prove a conjecture about a specific type of 3D space.
problem Guilloux's conjecture about the Borel function on a hyperbolic 3-manifold.
method Proved Guilloux's conjecture for a particular reflection group.
result The Borel function is rigid at infinity for the tetrahedral reflection lattice.
Unified 3D R-matrices from quantum cluster algebra.
problem Constructing new solutions to the tetrahedron equation.
method Symmetric butterfly quiver, quantum cluster algebra, quantum dilogarithms, q-Weyl algebra.
result Unified 3D R-matrices from various sources.
Minimal surfaces span periodic curves in 3D space.
problem Existence of minimal surfaces spanning periodic curves.
method Proof of existence for minimal surfaces using periodic curves in R3. result Existence of noncompact simply connected periodic minimal surfaces.
The paper proves a Minkowski-like theorem for tetrahedra in dS3 and AdS3.
problem Formulating and proving a constant-curvature, holonomy-valued Lorentzian analogue of Minkowski theorem for tetrahedra.
method Formulated and proved a Lorentzian analogue of Minkowski theorem for tetrahedra in dS3 and AdS3.
result A unique strictly convex tetrahedron can be reconstructed from four non-trivial based SO+(1,2) holonomies.
We identify a duplicate pair in the well-known Callahan-Hildebrand-Weeks census of cusped finite-volume hyperbolic 3-manifolds. Specifically, the six-tetrahedron non-orientable manifolds x101 and x103 are homeomorphic.
We address the question of determining the eigenvalues λ_n (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 n nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), …
The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum volume conjecture) are extended to $U_q(\fraksl_2)$ colored quantum invariants of the theta and tetrahedron graph. The $\SL(2,\bC)$ character variety of the fundamental group of the complement of a trivalent graph with E edges in S3 is a L…
Proves Seidel's conjectures about ideal tetrahedra in hyperbolic 3-space.
problem Determining the volume of ideal hyperbolic tetrahedra using algebraic maps.
method Analyzes the doubly stochastic Gram matrix of tetrahedron vertices.
result Volume is a monotonic function of the permanent and determinant of the Gram matrix.
We give a method to construct non symmetric solutions of a global tetrahedron equation from solutions of the Yang-Baxter equation. The solution in the HOMFLYPT case gives rise to the first combinatorial quantum 1-cocycle which represents a non trivial cohomology class in the topological moduli space of long knots. We c…
We show that a topologically minimal disk in a tetrahedron with index n is either a normal triangle, a normal quadrilateral, or a normal helicoid with boundary length 4(n+1). This mirrors geometric results of Colding and Minicozzi.
Study complex reflections in infinite Coxeter tetrahedron moduli space.
problem Characterize representations of Coxeter group in complex hyperbolic space.
method Type-preserving representations of Coxeter group G to PU(3,1), parameterized by θ. result Discrete and faithful representations for θ∈[65π,π]. First nontrivial moduli space in complex hyperbolic space. Study proves properties of compact Hermitian surfaces with specific curvature conditions.
problem Characterizing compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature.
method Analyzes surfaces with Gauduchon connections and Lichnerowicz holomorphic sectional curvature.
result Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature are either Kähler or isosceles Hopf surfaces.
I follow Y. Yokota to explain how to obtain a tetrahedron decomposition of the complement of a hyperbolic knot and compare it with the asymptotic behavior of Kashaev's link invariant using the figure-eight knot as an example.
Study integrability of quantized six-vertex model on torus.
problem Integrability of a specific lattice model on a torus.
method Defined layer transfer matrices and tetrahedron equations for admissible graphs.
result Established commutativity of transfer matrices and derived quantum Hamiltonians.
We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index n. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a particular normal form with respect to any triangulation.
Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.
problem Parameterize spectral data of constant mean curvature tori in R^3.
method Use Whitham deformations, blowups, and spectral data analysis.
result Prove the Wente family is parameterized by the bisector of the right angle.