On correlation of hyperbolic volumes of fullerenes with their properties

We observe that fullerene graphs are one-skeletons of polyhedra, which can be realized with all dihedral angles equal to π /2 in a hyperbolic 3-dimensional space. One of the most important invariants of such a polyhedron is its volume. We are referring this volume as a hyperbolic volume of a fullerene. It is known that some topological indices of graphs of chemical compounds serve as strong descriptors and correlate with chemical properties. We demonstrate that hyperbolic volume of fullerenes correlates with few important topological indices and so, hyperbolic volume can serve as a chemical descriptor too. The correlation between hyperbolic volume of fullerene and its Wiener index suggested few conjectures on volumes of hyperbolic polyhedra. These conjectures are confirmed for the initial list of fullerenes.

Yokota type invariants derived from non-integral highest weight representations of 𝒰q(sl2)

We define invariants for colored oriented spatial graphs by generalizing CM invariants [F. Costantino and J. Murakami, On [Formula: see text] quantum [Formula: see text]-symbols and their relation to the hyperbolic volume, Quantum Topol. 4 (2013) 303–351], which were defined via non-integral highest weight representations of [Formula: see text]. We apply the same method used to define Yokota’s invariants, and we call these invariants Yokota type invariants. Then, we propose a volume conjecture of the Yokota type invariants of plane graphs, which relates to volumes of hyperbolic polyhedra corresponding to the graphs, and check it numerically for some square pyramids and pentagonal pyramids.

Isometric embeddings of polyhedra into Euclidean space

In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space [Formula: see text] which admits a triangulation [Formula: see text] such that each n-dimensional simplex of [Formula: see text] is affinely isometric to a simplex in 𝔼n. We prove that any 1-Lipschitz map from an n-dimensional Euclidean polyhedron [Formula: see text] into 𝔼3n is ϵ-close to a pl isometric embedding for any ϵ > 0. If we remove the condition that the map be pl, then any 1-Lipschitz map into 𝔼2n + 1 can be approximated by a (continuous) isometric embedding. These results are extended to isometric embedding theorems of spherical and hyperbolic polyhedra into Euclidean space by the use of the Nash–Kuiper C1 isometric embedding theorem ([9] and [13]).