664 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
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
Cluster algebras of type D: pseudotriangulations approach
We present a combinatorial model for cluster algebras of type in terms
of centrally symmetric pseudotriangulations of a regular -gon with a small
disk in the centre. This model provides convenient and uniform interpretations
for clusters, cluster variables and their exchange relations, as well as for
quivers and their mutations. We also present a new combinatorial interpretation
of cluster variables in terms of perfect matchings of a graph after deleting
two of its vertices. This interpretation differs from known interpretations in
the literature. Its main feature, in contrast with other interpretations, is
that for a fixed initial cluster seed, one or two graphs serve for the
computation of all cluster variables. Finally, we discuss applications of our
model to polytopal realizations of type associahedra and connections to
subword complexes and -cluster complexes.Comment: 21 pages, 21 figure
Symmetric matrices, Catalan paths, and correlations
Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix
in terms of connected principal and almost-principal minors. Each entry is an
explicit Laurent polynomial whose terms are the weights of domino tilings of a
half Aztec diamond. They conjectured an analogue of this parametrization for
symmetric matrices, where the Laurent monomials are indexed by Catalan paths.
In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a
statistics problem pioneered by Joe (2006). Correlation matrices are
represented by an explicit bijection from the cube to the elliptope
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