3,798 research outputs found
On signed diagonal flip sequences
Eliahou \cite{2} and Kryuchkov \cite{9} conjectured a proposition that
Gravier and Payan \cite{4} proved to be equivalent to the Four Color Theorem.
It states that any triangulation of a polygon can be transformed into another
triangulation of the same polygon by a sequence of signed diagonal flips. It is
well known that any pair of polygonal triangulations are connected by a
sequence of (non-signed) diagonal flips. In this paper we give a sufficient and
necessary condition for a diagonal flip sequence to be a signed diagonal flip
sequence.Comment: 11 pages, 24 figures, to appear in European Journal of Combinatoric
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
On the resonance eigenstates of an open quantum baker map
We study the resonance eigenstates of a particular quantization of the open
baker map. For any admissible value of Planck's constant, the corresponding
quantum map is a subunitary matrix, and the nonzero component of its spectrum
is contained inside an annulus in the complex plane, . We consider semiclassical sequences of eigenstates, such that the
moduli of their eigenvalues converge to a fixed radius . We prove that, if
the moduli converge to , then the sequence of eigenstates
converges to a fixed phase space measure . The same holds for
sequences with eigenvalue moduli converging to , with a different
limit measure . Both these limiting measures are supported on
fractal sets, which are trapped sets of the classical dynamics. For a general
radius , we identify families of eigenstates with
precise self-similar properties.Comment: 32 pages, 2 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
Associahedra via spines
An associahedron is a polytope whose vertices correspond to triangulations of
a convex polygon and whose edges correspond to flips between them. Using
labeled polygons, C. Hohlweg and C. Lange constructed various realizations of
the associahedron with relevant properties related to the symmetric group and
the classical permutahedron. We introduce the spine of a triangulation as its
dual tree together with a labeling and an orientation. This notion extends the
classical understanding of the associahedron via binary trees, introduces a new
perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's
original approach, and sheds light upon the combinatorial and geometric
properties of the resulting realizations of the associahedron. It also leads to
noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction
- …