arXiv research

Proves volume conjecture for double twist knots using complexified tetrahedrons.

First working model of a monostable tetrahedron described.

New Bailey pairs derived for tetrahedron index, linking knot invariants.

Quantum theory of curved tetrahedrons yields quantum group intertwiners.

Study special complexified Kähler forms in mirror symmetry.

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.

Extends K-energy to complexified Kähler classes for scalar curvature study.

Quantum $6j$-symbols linked to tetrahedra angles and volumes.

Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.

Knots can be embedded into fractals like the Menger Sponge and Sierpinski Tetrahedron.

Complex $b$-$6j$ symbols relate to hyperbolic 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…

By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt…

Simple geodesics on spherical tetrahedra identified for specific angles.

Develops quantum cluster algebra approach to solve tetrahedron equation.

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…

We shall construct a natural Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of the flat-ness of our Higgs bundle, we find a commutator identity that ca…

New combinatorial model for Milnor fibration using oriented matroids.

Machine learning finds Z/2 eigenfunctions on a sphere.

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…

Asymptotics of quantum $6j$ symbols corresponding to a hyperbolic tetrahedra is investigated and the first two leading terms are determined for the case that the tetrahedron has a ideal or ultra-ideal vertex. These terms are given by the volume and the determinant of the Gram matrix of the tetrahedron. A relation to th…

The set of maximal non-integrable structures $(SU(2)\times SU(2),B,I)$, where $B$ is Killing-Cartan metric is described as subset of $\mathbb{CP}^3$. The visualization of complex projective space $\mathbb{CP}^3$ as tetrahedron which edges and faces are $\mathbb{CP}^1$ and $\mathbb{CP}^2$ is used.

The paper constructs homotopically non-trivial spheres in complexified spaces.

Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.

It is known that there exist Calabi-Yau structures on the complexifications of symmetric spaces of compact type. In this paper, we describe the Calabi-Yau structures of the complexified symmetric spaces in terms of the Schwarz's theorem in detail. We consider the case where the Calabi-Yau structure arises from the Riem…

We study the Kaehler metric given by the logarithm of a cubic form on its complexified index cone. Under mirror symmetry, this metric should asymptotically correspond to the Weil-Petersson metric. Using the theory of special Kaehler manifolds, a proof of a curvature formula for this metric is given.

Models for 3D harmonic 1-forms and spinors near singular points.

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.

The paper describes orbits of parabolic subgroups in complexified actions.

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 $8$-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 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…

Constructs simplified or complexified simplicial complexes.

Let $V$ be a finite dimensional complex vector space and $W\subseteq \GL(V)$ be a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$ of the reflecting hyperplanes. We prove that $V^{\reg}$ is a $K(π,1)$ space. This was predicted by a classical conjecture, originally stated by Brieskorn for complex…

Unified 3D R-matrices from quantum cluster algebra.

Subharmonicity of Dirichlet energy proven for Kähler manifolds.

Minimal surfaces span periodic curves in 3D space.

We prove, in the case of hyperbolic 3-space, a couple of conjectures raised by J. J. Seidel in "On the volume of a hyperbolic simplex", Stud. Sci. Math. Hung. 21, 243-249, 1986. These conjectures concern expressing the volume of an ideal hyperbolic tetrahedron as a monotonic function of algebraic maps. More precisely, …

The paper proves a Minkowski-like theorem for tetrahedra in dS3 and AdS3.

Adds layers to NNs to protect them from reverse engineering.

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.

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 $S^3$ is a L…

Let $V$ be a finite dimensional complex vector space and $W\subset \GL(V)$ be a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$ of the reflecting hyperplanes. A classical conjecture predicts that $V^{\reg}$ is a $K(pi,1)$ space. When $W$ is a complexified real reflection group, the conjecture f…

The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.

Study of complexified Hermitian symmetric spaces and their structures.

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.

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.

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…