127 research outputs found
A History of Flips in Combinatorial Triangulations
Given two combinatorial triangulations, how many edge flips are necessary and
sufficient to convert one into the other? This question has occupied
researchers for over 75 years. We provide a comprehensive survey, including
full proofs, of the various attempts to answer it.Comment: Added a paragraph referencing earlier work in the vertex-labelled
setting that has implications for the unlabeled settin
Irreducible Triangulations are Small
A triangulation of a surface is \emph{irreducible} if there is no edge whose
contraction produces another triangulation of the surface. We prove that every
irreducible triangulation of a surface with Euler genus has at most
vertices. The best previous bound was .Comment: v2: Referees' comments incorporate
Diagonal Flips of Triangulations on Closed Surfaces Preserving Specified Properties
AbstractConsider a class P of triangulations on a closed surfaceF2, closed under vertex splitting. We shall show that any two triangulations with the same and sufficiently large number of vertices which belong to P can be transformed into each other, up to homeomorphism, by a finite sequence of diagonal flips through P. Moreover, if P is closed under homeomorphism, then the condition “up to homeomorphism” can be replaced with “up to isotopy.
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
Multi-triangulations as complexes of star polygons
Maximal -crossing-free graphs on a planar point set in convex
position, that is, -triangulations, have received attention in recent
literature, with motivation coming from several interpretations of them.
We introduce a new way of looking at -triangulations, namely as complexes
of star polygons. With this tool we give new, direct, proofs of the fundamental
properties of -triangulations, as well as some new results. This
interpretation also opens-up new avenues of research, that we briefly explore
in the last section.Comment: 40 pages, 24 figures; added references, update Section
Triangulating the Real Projective Plane
We consider the problem of computing a triangulation of the real projective
plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a
triangulation of P2 always exists if at least six points in S are in general
position, i.e., no three of them are collinear. We also design an algorithm for
triangulating P2 if this necessary condition holds. As far as we know, this is
the first computational result on the real projective plane
- …