arXiv research

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

We obtain a nature generalization for an affine Sierpinski carpet and Sierpinski triangle to $n$-dimensional space, by using the generations and characterizations of affinely-equivalent Sierpinski carpet. Exactly, in this paper, a Menger sponge and Sierpinski simplex in $4$-dimensional space could be drawn out clearly …

The paper shows how certain groups' boundaries relate to the Sierpiński carpet.

Uniform bounds found for Sierpinski carpet hyperbolic components.

We give a necessary and sufficient condition for a hyperbolic Coxeter group with planar nerve to have Sierpiński curve as its Gromov boundary.

Characterizes Coxeter groups with specific boundary shapes.

For arbitrary integer n, we describe a large class of right-angled Coxeter systems for which the visual baundary (of the corresponding Coxeter-Davis complex) is homeomorphic to the n-dimensional Sierpiński compactum. We also provide a necessary and sufficient condition for a planar simplicial complex L under which the …

For n>6, we show that if G is a torsion-free hyperbolic group whose visual boundary is an (n-2)-dimensional Sierpinski space, then G=π_1(W) for some aspherical n-manifold W with nonempty boundary. Concerning the converse, we construct, for each n>3, examples of aspherical manifolds with boundary, whose fundamental grou…

We study a new class of square Sierpiński carpets $F_{n,p}$ ($5\leq n, 1\leq p<\frac{n}{2}-1$) on $\mathbb{S}^2$, which are not quasisymmetrically equivalent to the standard Sierpiński carpets. We prove that the group of quasisymmetric self-maps of each $F_{n,p}$ is the Euclidean isometry group. We also establish that …

The paper extends Hodge-de Rham theory to higher-dimensional Sierpinski gaskets.

First working model of a monostable tetrahedron described.

Proves volume conjecture for double twist knots using complexified tetrahedrons.

This paper constructs a continuous decomposition of the Sierpiński curve into acyclic continua one of which is an arc. This decomposition is then used to construct another continuous decomposition of the Sierpiński curve. The resulting decomposition space is homeomorphic to the continuum obtained from taking the Sierpi…

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

Classifies horocycle flow closures in hyperbolic 3-manifolds.

Quantum theory of curved tetrahedrons yields quantum group intertwiners.

A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poinca…

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.

If a torsion-free hyperbolic group G has 1-dimensional boundary, then the boundary is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When the boundary of G is a Sierpinski carpet we show that G is a quasi-convex subgroup of a 3-dimensional hyperbolic Poincare duality group. We also…

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

Researchers create a Fredholm module on fractal shapes like the Cantor set.

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…

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve …

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…

J. Kigami has laid the foundations of what is now known as analysis on fractals, by allowing the construction of an operator of the same nature of the Laplacian, defined locally, on graphs having a fractal character. The Sierpinski gasket stands out of the best known example. It has, since then, been taken up, develope…

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.

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

The paper examines properties of self-affine Sierpiński sponges using metric invariants.

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on $1$-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have pos…

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.

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…

This paper classifies fundamental groups of perforated surfaces.

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…

We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic $3$ manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schott…

Unified 3D R-matrices from quantum cluster algebra.

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.

A theorem controls the relationship between dimensions of continua.