Algorithmic construction and recognition of hyperbolic 3-manifolds, links, and graphs

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric boundary (up to graph complexity 12) and for infinite families of lens spaces. In this paper we extend to graph complexity 14 the computations for orientable ma...

This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers basic tools in hyperbolic geometry and geometric structures on 3-manifolds. The second part focuses on families of knots and links that have been amenable to study via hyperbolic geometry, particularly twist knots, 2-bridge knots, and alternating knots. It also develops geometric...

This article presents a survey of some recent results in the theory of spatial graphs. In particular, we highlight results related to intrinsic knotting and linking and results about symmetries of spatial graphs. In both cases we consider spatial graphs in $S^3$ as well as in other $3$-manifolds.

This is the first in a series of papers showing that Haken manifolds have hyperbolic structures; this first was published, the second two have existed only in preprint form, and later preprints were never completed. This eprint is only an approximation to the published version, which is the definitive form for part I, and is provided for convenience only. All references and quotations should be taken from the published version, since the theorem numbering is different and not...

We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.

We provide an algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary. More generally, we supply an algorithm to determine (up to ambient isotopy) all the Heegaard splittings of any given genus for the manifold. As a consequence, the tunnel number of a hyperbolic link is algorithmically computable. Our techniques rely on Rubinstein's work on almost normal surfaces, and also a new structure called a partially flat angled ideal triangulation.

In this paper we present a classical construction of the Hyperbolic structure of the complement of a link in the sense of Thurston for the particular case of the Borromean rings link. As this is nothing new, the aim of this paper is to complete the literature about this topic in the particular case of the Borromean rings complement.

In this paper we are interested in computing representations of the fundamental group of a 3-manifold into PSL(3;C) (in particular in PSL(2;C); PSL(3;R) and PU(2; 1)). The representations are obtained by gluing decorated tetrahedra of flags. We list complete computations (giving 0-dimensional or 1-dimensional solution sets) for the first complete hyperbolic non-compact manifolds with finite volume which are obtained gluing less than three tetrahedra with a description of the ...

This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on three-dimensional surfaces. The survey focuses on polyhedral three-dimensional surfaces.

This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have ...