arXiv research

Trees and their products compared to octahedron.

We use the consistency approach to classify discrete integrable 3D equations of the octahedron type. They are naturally treated on the root lattice $Q(A_3)$ and are consistent on the multidimensional lattice $Q(A_N)$. Our list includes the most prominent representatives of this class, the discrete KP equation and its S…

Mazur's knot exterior is described by a single regular ideal octahedron, leading to hyperbolic structures related to the Whitehead link.

The main results of A. Zorich and I. Dynnikov about plane sections of periodic surfaces are extended to the PL case. As an application, the Stereographic Map of a truncated octahedron, extended to the whole $\Rt$ by periodicity, is analyzed numerically.

Proves certain alternating links have specific geometric properties.

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

Study smoothings of singular intersections of ellipsoids.

We try to give a cluster algebraic interpretation of complex volume of knots. We construct the R-operator from the cluster mutations, and we show that it is regarded as a hyperbolic octahedron. The cluster variables are interpreted as edge parameters used by Zickert in computing complex volume.

In this article, we describe symplectic and complex toric spaces associated to the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational and the regular icosahedron is neither simple nor rational. We re…

The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We…

Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central role played by hyperbolic geometry within three-dimensional topology. We will also point out the striking difference wi…

We prove that the 8^4_2 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V_8 of the ideal regular octahedron, i.e. 7.32... = 2V_8. The proof relies on Agol's argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also nee…

The $\textit{volume density}$ of a hyperbolic link $K$ is defined to be the ratio of the hyperbolic volume of $K$ to the crossing number of $K$. We show that there are sequences of non-alternating links with volume density approaching $v_8$, where $v_8$ is the volume of the ideal hyperbolic octahedron. We show that the…

The study classifies tilings of the sphere by congruent quadrilaterals.

We establish a correspondence between the dimer model on a bipartite graph and a circle pattern with the combinatorics of that graph, which holds for graphs that are either planar or embedded on the torus. The set of positive face weights on the graph gives a set of global coordinates on the space of circle patterns wi…

We prove a rigidity theorem for the geometry of the unit ball in random subspaces of the scl norm in B_1^H of a free group. In a free group F of rank k, a random word w of length n (conditioned to lie in [F,F]) has scl(w)=log(2k-1)n/6log(n) + o(n/log(n)) with high probability, and the unit ball in a subspace spanned by…

We enumerate all spaces obtained by gluing in pairs the faces of the octahedron in an orientation-reversing fashion. Whenever such a gluing gives rise to non-manifold points, we remove small open neighbourhoods of these points, so we actually deal with three-dimensional manifolds with (possibly empty) boundary. There a…

We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichmüller curves. Using an algorithm that can be used generally to compute Teichmüller curves of translation covers of p…

We present explicit geometric decompositions of the complement of tiling links, which are alternating links whose projection graphs are uniform tilings of the 2-sphere, the Euclidean plane or the hyperbolic plane. This requires generalizing the angle structures program of Casson and Rivin for triangulations with a mixt…

The study broadens the concept of cyclic polytopes to Veronese polytopes.

We present explicit geometric decompositions of the hyperbolic complements of alternating $k$-uniform tiling links, which are alternating links whose projection graphs are $k$-uniform tilings of $S^2$, $\mathbb{E}^2$, or $\mathbb{H}^2$. A consequence of this decomposition is that the volumes of spherical alternating $k…

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.…

Self-affine tiles homeomorphic to a ball proven for a specific digit set.

Improved lower bounds on volumes of hyperbolic 3-manifolds with specific topologies.

Let $X\in\mathbb{R}^{n}$. For $φ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $φ^{t}=t^{-1}φ(Xt)$. A projective flow is a solution to the projective translation equation $φ^{t+s}=φ^{t}\circφ^{s}$, $t,s\in\mathbb{R}$. The projective superflow is a projective flow with a rational vector field which, …

Let $x\in\mathbb{R}^{n}$. For $φ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $φ^{t}=t^{-1}φ(xt)$. A projective flow is a solution to the projective translation equation $φ^{t+s}=φ^{t}\circφ^{s}$, $t,s\in\mathbb{R}$. Previously we have developed an arithmetic, topologic and analytic theory of $2$-d…

The study examines the normalized volumes of right-angled hyperbolic polyhedra and their spectra.