273 research outputs found
Generating families of surface triangulations. The case of punctured surfaces with inner degree at least 4
We present two versions of a method for generating all triangulations of any
punctured surface in each of these two families: (1) triangulations with inner
vertices of degree at least 4 and boundary vertices of degree at least 3 and
(2) triangulations with all vertices of degree at least 4. The method is based
on a series of reversible operations, termed reductions, which lead to a
minimal set of triangulations in each family. Throughout the process the
triangulations remain within the corresponding family. Moreover, for the family
(1) these operations reduce to the well-known edge contractions and removals of
octahedra. The main results are proved by an exhaustive analysis of all
possible local configurations which admit a reduction.Comment: This work has been partially supported by PAI FQM-164; PAI FQM-189;
MTM 2010-2044
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
Surface cubications mod flips
Let be a compact surface. We prove that the set of surface
cubications modulo flips, up to isotopy, is in one-to-one correspondence with
.Comment: revised version, 18
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
The geometry of flip graphs and mapping class groups
The space of topological decompositions into triangulations of a surface has
a natural graph structure where two triangulations share an edge if they are
related by a so-called flip. This space is a sort of combinatorial
Teichm\"uller space and is quasi-isometric to the underlying mapping class
group. We study this space in two main directions. We first show that strata
corresponding to triangulations containing a same multiarc are strongly convex
within the whole space and use this result to deduce properties about the
mapping class group. We then focus on the quotient of this space by the mapping
class group to obtain a type of combinatorial moduli space. In particular, we
are able to identity how the diameters of the resulting spaces grow in terms of
the complexity of the underlying surfaces.Comment: 46 pages, 23 figure
Unimodular lattice triangulations as small-world and scale-free random graphs
Real-world networks, e.g. the social relations or world-wide-web graphs,
exhibit both small-world and scale-free behaviour. We interpret lattice
triangulations as planar graphs by identifying triangulation vertices with
graph nodes and one-dimensional simplices with edges. Since these
triangulations are ergodic with respect to a certain Pachner flip, applying
different Monte-Carlo simulations enables us to calculate average properties of
random triangulations, as well as canonical ensemble averages using an energy
functional that is approximately the variance of the degree distribution. All
considered triangulations have clustering coefficients comparable with real
world graphs, for the canonical ensemble there are inverse temperatures with
small shortest path length independent of system size. Tuning the inverse
temperature to a quasi-critical value leads to an indication of scale-free
behaviour for degrees . Using triangulations as a random graph model
can improve the understanding of real-world networks, especially if the actual
distance of the embedded nodes becomes important.Comment: 17 pages, 6 figures, will appear in New J. Phy
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.
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