216 research outputs found
On canonical triangulations of once-punctured torus bundles and two-bridge link complements
We prove the hyperbolization theorem for punctured torus bundles and
two-bridge link complements by decomposing them into ideal tetrahedra which are
then given hyperbolic structures, following Rivin's volume maximization
principle.Comment: This is the version published by Geometry & Topology on 16 September
2006. Appendix by David Fute
Angled decompositions of arborescent link complements
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main application is a new proof of a classical, unpublished theorem of Bonahon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 hyperbolic complements.Comment: 42 pages, 23 figures. Slightly expanded exposition and reference
One brick at a time: a survey of inductive constructions in rigidity theory
We present a survey of results concerning the use of inductive constructions
to study the rigidity of frameworks. By inductive constructions we mean simple
graph moves which can be shown to preserve the rigidity of the corresponding
framework. We describe a number of cases in which characterisations of rigidity
were proved by inductive constructions. That is, by identifying recursive
operations that preserved rigidity and proving that these operations were
sufficient to generate all such frameworks. We also outline the use of
inductive constructions in some recent areas of particularly active interest,
namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar
frameworks. We summarize the key outstanding open problems related to
inductions.Comment: 24 pages, 12 figures, final versio
Topological Phases: An Expedition off Lattice
Motivated by the goal to give the simplest possible microscopic foundation
for a broad class of topological phases, we study quantum mechanical lattice
models where the topology of the lattice is one of the dynamical variables.
However, a fluctuating geometry can remove the separation between the system
size and the range of local interactions, which is important for topological
protection and ultimately the stability of a topological phase. In particular,
it can open the door to a pathology, which has been studied in the context of
quantum gravity and goes by the name of `baby universe', Here we discuss three
distinct approaches to suppressing these pathological fluctuations. We
complement this discussion by applying Cheeger's theory relating the geometry
of manifolds to their vibrational modes to study the spectra of Hamiltonians.
In particular, we present a detailed study of the statistical properties of
loop gas and string net models on fluctuating lattices, both analytically and
numerically.Comment: 38 pages, 22 figure
Explicit angle structures for veering triangulations
Agol recently introduced the notion of a veering triangulation, and showed
that such triangulations naturally arise as layered triangulations of fibered
hyperbolic 3-manifolds. We prove, by a constructive argument, that every
veering triangulation admits positive angle structures, recovering a result of
Hodgson, Rubinstein, Segerman, and Tillmann. Our construction leads to explicit
lower bounds on the smallest angle in this positive angle structure, and to
information about angled holonomy of the boundary tori.Comment: 23 pages, 8 figures. v2 contains a cleaner definition of holonomy in
Section 6.1, and minor expository changes throughout. To appear in Algebraic
& Geometric Topolog
Cusp areas of Farey manifolds and applications to knot theory
This paper gives the first explicit, two-sided estimates on the cusp area of
once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link
complements. The input for these estimates is purely combinatorial data coming
from the Farey tesselation of the hyperbolic plane. The bounds on cusp area
lead to explicit bounds on the volume of Dehn fillings of these manifolds, for
example sharp bounds on volumes of hyperbolic closed 3-braids in terms of the
Schreier normal form of the associated braid word. Finally, these results are
applied to derive relations between the Jones polynomial and the volume of
hyperbolic knots, and to disprove a related conjecture.Comment: 44 pages, 11 figures. Version 4 contains revisions and corrections
(most notably, in Sections 5 and 6) that incorporate referee comments. To
appear in the International Mathematics Research Notices
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