Ideal Hyperbolic Polyhedra and Discrete Uniformization

We study a subclass of alternating links for which the complete hyperbolic metric can be realised directly by pairwise identification of faces of two ideal hyperbolic polyhedra. Our main result is a characterization of these links: essentially, the corresponding polyhedra are exactly the Archimedean solids with trivalent vertices. Furthermore, we show that the only knots which arise are the two dodecahedral knots, and the figure eight knot.

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Whitehead groups of certain hyperbolic manifolds

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100061557 ◽

1984 ◽

Vol 95 (2) ◽

pp. 299-308 ◽

Cited By ~ 1

Author(s):

A. J. Nicas ◽

C. W. Stark

Keyword(s):

Class Group ◽

Universal Cover ◽

Hyperbolic Manifolds ◽

Fundamental Groups ◽

Whitehead Group ◽

Free Products ◽

Projective Class ◽

Aspherical Manifold ◽

Hyperbolic Polyhedra ◽

Whitehead Groups

An aspherical manifold is a connected manifold whose universal cover is contractible. It has been conjectured that the Whitehead groups Whj (π1 M) (including the projective class group, the original Whitehead group of π1M, and the higher Whitehead groups of [9]) vanish for any compact aspherical manifold M. The present paper considers this conjecture for twelve hyperbolic 3-manifolds constructed from regular hyperbolic polyhedra. Hyperbolic manifolds are of special interest in this regard since so much is known about their topology and geometry and very little is known about the algebraic K-theory of hyperbolic manifolds whose fundamental groups are not generalized free products.

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

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216516500541 ◽

2016 ◽

Vol 25 (10) ◽

pp. 1650054

Author(s):

Atsuhiko Mizusawa ◽

Jun Murakami

Keyword(s):

Plane Graphs ◽

Highest Weight ◽

Volume Conjecture ◽

Hyperbolic Volume ◽

Spatial Graphs ◽

Hyperbolic Polyhedra ◽

Highest Weight Representations

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.

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On an Elementary Proof of Rivin's Characterization of Convex Ideal Hyperbolic Polyhedra by their Dihedral Angles

Geometriae Dedicata ◽

10.1007/s10711-004-3180-y ◽

2004 ◽

Vol 108 (1) ◽

pp. 111-124 ◽

Cited By ~ 2

Author(s):

François Guéritaud

Keyword(s):

Elementary Proof ◽

Dihedral Angles ◽

Hyperbolic Polyhedra

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On correlation of hyperbolic volumes of fullerenes with their properties

Computational and Mathematical Biophysics ◽

10.1515/cmb-2020-0108 ◽

2020 ◽

Vol 8 (1) ◽

pp. 150-167

Author(s):

A. A. Egorov ◽

A Yu. Vesnin

Keyword(s):

Dimensional Space ◽

Chemical Properties ◽

Wiener Index ◽

Topological Indices ◽

3 Dimensional ◽

Hyperbolic Volume ◽

Chemical Descriptor ◽

Hyperbolic Polyhedra ◽

Hyperbolic Volumes ◽

Initial List

AbstractWe 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.

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