54 research outputs found
Non-geometric veering triangulations
Recently, Ian Agol introduced a class of "veering" ideal triangulations for
mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the
singular points. These triangulations have very special combinatorial
properties, and Agol asked if these are "geometric", i.e. realised in the
complete hyperbolic metric with all tetrahedra positively oriented. This paper
describes a computer program Veering, building on the program Trains by Toby
Hall, for generating these triangulations starting from a description of the
homeomorphism as a product of Dehn twists. Using this we obtain the first
examples of non-geometric veering triangulations; the smallest example we have
found is a triangulation with 13 tetrahedra
Veering triangulations admit strict angle structures
Agol recently introduced the concept of a veering taut triangulation, which
is a taut triangulation with some extra combinatorial structure. We define the
weaker notion of a "veering triangulation" and use it to show that all veering
triangulations admit strict angle structures. We also answer a question of
Agol, giving an example of a veering taut triangulation that is not layered.Comment: 15 pages, 9 figure
Non ambiguous structures on 3-manifolds and quantum symmetry defects
The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic
oriented cusped -manifolds can be split in a "symmetrization" factor and a
"reduced" state sum. We show that these factors are invariants on their own,
that we call "symmetry defects" and "reduced QHI", provided the manifolds are
endowed with an additional "non ambiguous structure", a new type of
combinatorial structure that we introduce in this paper. A suitably normalized
version of the symmetry defects applies to compact -manifolds endowed with
-characters, beyond the case of cusped manifolds. Given a
manifold with non empty boundary, we provide a partial "holographic"
description of the non-ambiguous structures in terms of the intrinsic geometric
topology of . Special instances of non ambiguous structures can be
defined by means of taut triangulations, and the symmetry defects have a
particularly nice behaviour on such "taut structures". Natural examples of taut
structures are carried by any mapping torus with punctured fibre of negative
Euler characteristic, or by sutured manifold hierarchies. For a cusped
hyperbolic -manifold which fibres over , we address the question of
determining whether the fibrations over a same fibered face of the Thurston
ball define the same taut structure. We describe a few examples in detail. In
particular, they show that the symmetry defects or the reduced QHI can
distinguish taut structures associated to different fibrations of . To
support the guess that all this is an instance of a general behaviour of state
sum invariants of 3-manifolds based on some theory of 6j-symbols, finally we
describe similar results about reduced Turaev-Viro invariants.Comment: 58 pages, 32 figures; exposition improved, ready for publicatio
Arbitrarily large veering triangulations with a vanishing taut polynomial
Landry, Minsky, and Taylor introduced an invariant of veering triangulations
called the taut polynomial. Via a connection between veering triangulations and
pseudo-Anosov flows, it generalizes the Teichm\"uller polynomial of a fibered
face of the Thurston norm ball to (some) non-fibered faces. We construct a
sequence of veering triangulations, with the number of tetrahedra tending to
infinity, whose taut polynomials vanish. These veering triangulations encode
non-circular Anosov flows transverse to tori.Comment: 26 pages, 12 figure
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