Journal ArticleDOI
Hyperbolic volumes of Fibonacci manifolds
A. Yu. Vesnin,Alexander Mednykh +1 more
TLDR
In this paper, Yu et al. studied three-dimensional compact orientable hyperbolic manifolds connected with the Fibonacci groups and proved that the group F(2, m) is finite if and only if m = 1,2,3, 4,5, 7.817.Abstract:
A. Yu. Vesnin and A. D. Mednykh UDC 515.16 + 512.817.7 This article is devoted to the study of three-dimensional compact orientable hyperbolic manifolds connected with the Fibonacci groups. The Fibonacci groups F(2, m) = (Zl, z2,..., z,n : ziZi+l = zi+2, { mod m) were introduced by J. Conway [1]. The first natural question connected with these groups was whether they are finite or not [1]. It is known from [2-6] that the group F(2, m) is finite if and only if m = 1,2,3, 4,5, 7. Some algebraic generalizations of the groupsread more
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All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials
Gaëtan Borot,Bertrand Eynard +1 more
TL;DR: In this paper, a conjecture was proposed to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, J_N(q = exp(2u/N)) when N goes to infinity.
Journal ArticleDOI
Two-sided asymptotic bounds for the complexity of some closed hyperbolic three-manifolds
TL;DR: In this paper, two-sided bounds for the complexity of two infinite series of closed orientable 3-dimensional hyperbolic manifolds, the Lobell manifold and the Fibonacci manifold, were established.
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Generalised Fibonacci manifolds
C. Maclachlan,Alan W. Reid +1 more
TL;DR: In this paper, a 3-manifolds are constructed using related sequences of Lucas numbers, their geometric structures determined, and a curious relationship between the homology and the invariant trace field examined.
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On manifold spines and cyclic presentations of groups
TL;DR: A survey of results and open problems on compact 3-manifolds which admit spines corresponding to cyclic presentations of groups can be found in this paper, where the authors also discuss questions concerning spines of knot manifolds and regular neighborhoods of homotopically PL embedded compacta.
Journal ArticleDOI
Algebraic methods for solution of polyhedra
TL;DR: A survey of results on calculating the volumes of polyhedra in terms of their metrics and combinatorial structures can be found in this paper, where a generalization of Heron's formula for the area of a triangle to polyhedras is presented.
References
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Book
Knots and Links
TL;DR: In this paper, the fundamental group of three-dimensional PL geometry Seifert surfaces Finite cyclic coverings and the torsion invariants Infinite cyclic covers and the Alexander invariant Matrix invariants 3-manifolds and surgery on links Foliations, branched covers, fibrations and so on.
Book
Introduction to Knot Theory
Richard H. Crowell,Ralph H. Fox +1 more
TL;DR: In this article, the authors define knots and knots polynomials, and present the notion of a knot polynomial and a group of knots, and prove the van Kampen theorem.
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Volumes of hyperbolic three-manifolds
Walter D. Neumann,Don Zagier +1 more
TL;DR: In this paper, the set of all possible volumes of hyperbolic 3-manifolds is known to be a well-ordered subset of the real numbers and is of considerable interest (for number theoretic aspects see, for instance, [2], [13] and [15] ).
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