arXiv research

A simpler edge-based discretization method without dual volumes.

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 present examples of metric spaces that are not Riemannian manifolds nor dimensionally homogeneous that satisfy the Tetrahedral Property. In spite of that, Euclidean cones over metric spaces with small diameter do not satisfy this property. We extend Sormani's Tetrahedral Property to a less restrictive property and p…

We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the…

The tetrahedral index connects to a q-Bessel function, revealing new mathematical techniques.

Proof confirms volume conjecture for a specific knot.

Study on hidden symmetries in Dehn fillings of tetrahedral links.

The paper calculates Veech groups for triangulable structures on the sphere.

Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorf…

We prove that the Kontsevich tetrahedral flow $\dot{\mathcal{P}} = \mathcal{Q}_{a:b} (\mathcal{P})$, the right-hand side of which is a linear combination of two differential monomials of degree four in a bi-vector $\mathcal{P}$ on an affine real Poisson manifold $N^n$, does infinitesimally preserve the space of Poisson…

This paper explores a particular statistical model on 6-valent graphs with special properties which turns out to be invariant with respect to certain Roseman moves if the graph is the singular point graph of a diagram of a 2-knot. The approach uses the technic of the tetrahedral complex cohomology. We emphasize that th…

We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.

Prove realizability of genus-0 branch data for tetrahedral coverings of the sphere.

We study SU(2) BPS monopoles with spectral curves of the form $η^3+χ(ζ^6+b ζ^3-1)=0$. Previous work has has established a countable family of solutions to Hitchin's constraint that $L^2$ was trivial on such a curve. Here we establish that the only curves of this family that yield BPS monopoles correspond to tetrahedral…

Researchers create topologically protected knots in a realizable system.

Roseman moves are seven types of local modification for surface-link diagrams in $3$-space which generate ambient isotopies of surface-links in $4$-space. In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence. For each diagram of an…

We consider polyhedra and 4-polytopes in Minkowski spacetime - in particular, null polyhedra with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces. We present the basic properties of several classes of null-faced 4-polytopes: 4-simplices, "tetrahedral diamonds" and 4-parallelotopes. We propose …

Explicit solutions to the Riemann-Hilbert problem will be found realising some irreducible non-rigid local systems. The relation to isomonodromy and the sixth Painleve equation will be described. Keywords: Riemann-Hilbert problem, Painleve equations, algebraic solutions, Heun equations, tetrahedral/octahedral group, tr…

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…

Finite specializations of a q-deformed modular group at roots of unity.

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…

The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.

If a hyperbolic 3-manifold M admits a reducible and a finite Dehn filling, the distance between the filling slopes is known to be 1. This has been proved recently by Boyer, Gordon and Zhang. The first example of a manifold with two such fillings was given by Boyer and Zhang. In this paper, we give examples of hyperboli…

Half grid diagrams prove every link can be represented by a special type of grid diagram.

Grid homology confirms the Upsilon invariant in knot theory.

GridPyM handles grid diagrams for knot theory.

If $Γ$ is the fundamental group of a complete finite volume hyperbolic $3$-manifold, Guilloux conjectured that the Borel function on the $\text{PSL}(n,\mathbb{C})$-character variety of $Γ$ should be rigid at infinity, that is it should stay bounded away from its maximum at ideal points. In this paper we prove Guilloux'…

Grid homology theory for spatial graphs extends skein sequence.

Extends knot invariant to filtered grid complexes.

New method finds grid diagrams for many fibered knots.

Grid homology properties for MOY graphs studied.

Higher nilpotent analogues of the $A-\infty$-structure are explicitly defined on arbitrary simplicial complexes, generalizing explicit construction of /hep-th/0704.2609. These structures are associated with the higher nilpotent differential $d_n$, satisfying $d_n^n =0$, which is naturally defined on triangulated manifo…

Grid homology invariant proved for lens space links.

Computations based on explicit 4-periodic resolutions are given for the cohomology of the finite groups G known to act freely on S^3, as well as the cohomology rings of the associated 3-manifolds (spherical space forms) M = S^3/G. Chain approximations to the diagonal are constructed, and explicit contracting homotopies…

New trading strategy beats traditional grid in crypto markets.

New method constructs moduli spaces of Lagrangian surfaces in CP^2 from grid diagrams.

Develops equivariant grid homology for strongly invertible knots.

Grid homology shows knot unknotting lower bound.

Computes homology of an obstruction chain complex in grid homology.

Observational data hints at a finite universe, with spherical manifolds such as the Poincare dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes…

SKI accelerates GP inference with sparse grids to handle higher dimensions.

New proof shows not all Salem numbers are growth rates of Coxeter groups.

The paper studies grid homology for spatial graphs and proves a Künneth formula for connected sums.

Hexagon grid patterns emerge from conformal isometry in grid cell neural networks.

Minimal grid diagrams for 12-crossing prime knots identified.

It will be shown that according to theorems of K. Menger, every neuron grid if identified with a curve is able to preserve the adopted qualitative structure of a data space. Furthermore, if this identification is made, the neuron grid structure can always be mapped to a subset of a universal neuron grid which is constr…

The paper analyzes how grid cells perform path integration and learns hexagon grid patterns.

Power grids are one of the most important components of infrastructure in today's world. Every nation is dependent on the security and stability of its own power grid to provide electricity to the households and industries. A malfunction of even a small part of a power grid can cause loss of productivity, revenue and i…