990 research outputs found
Transforming triangulations on non planar-surfaces
We consider whether any two triangulations of a polygon or a point set on a
non-planar surface with a given metric can be transformed into each other by a
sequence of edge flips. The answer is negative in general with some remarkable
exceptions, such as polygons on the cylinder, and on the flat torus, and
certain configurations of points on the cylinder.Comment: 19 pages, 17 figures. This version has been accepted in the SIAM
Journal on Discrete Mathematics. Keywords: Graph of triangulations,
triangulations on surfaces, triangulations of polygons, edge fli
Flip-graph moduli spaces of filling surfaces
This paper is about the geometry of flip-graphs associated to triangulations
of surfaces. More precisely, we consider a topological surface with a
privileged boundary curve and study the spaces of its triangulations with n
vertices on the boundary curve. The surfaces we consider topologically fill
this boundary curve so we call them filling surfaces. The associated
flip-graphs are infinite whenever the mapping class group of the surface (the
group of self-homeomorphisms up to isotopy) is infinite, and we can obtain
moduli spaces of flip-graphs by considering the flip-graphs up to the action of
the mapping class group. This always results in finite graphs and we are
interested in their geometry.
Our main focus is on the diameter growth of these graphs as n increases. We
obtain general estimates that hold for all topological types of filling
surface. We find more precise estimates for certain families of filling
surfaces and obtain asymptotic growth results for several of them. In
particular, we find the exact diameter of modular flip-graphs when the filling
surface is a cylinder with a single vertex on the non-privileged boundary
curve.Comment: 52 pages, 29 figure
Once punctured disks, non-convex polygons, and pointihedra
We explore several families of flip-graphs, all related to polygons or
punctured polygons. In particular, we consider the topological flip-graphs of
once-punctured polygons which, in turn, contain all possible geometric
flip-graphs of polygons with a marked point as embedded sub-graphs. Our main
focus is on the geometric properties of these graphs and how they relate to one
another. In particular, we show that the embeddings between them are strongly
convex (or, said otherwise, totally geodesic). We also find bounds on the
diameters of these graphs, sometimes using the strongly convex embeddings.
Finally, we show how these graphs relate to different polytopes, namely type D
associahedra and a family of secondary polytopes which we call pointihedra.Comment: 24 pages, 6 figure
On ideal triangulations of surfaces up to branched transit equivalences
We consider triangulations of closed surfaces S with a given set of vertices
V; every triangulation can be branched that is enhanced to a Delta-complex.
Branched triangulations are considered up to the b-transit equivalence
generated by b-flips (i.e. branched diagonal exchanges) and isotopy keeping V
point-wise fixed. We extend a well known connectivity result for `naked'
triangulations; in particular in the generic case when S has negative
Euler-Poincare' characteristic c(S), we show that branched triangulations are
equivalent to each other if c(S) is even, while this holds also for odd c(S)
possibly after the complete inversion of one of the two branchings. Moreover we
show that under a mild assumption, two branchings on a same triangulation are
connected via a sequence of inversions of ambiguous edges (and possibly the
total inversion of one of them). A natural organization of the b-flips in
subfamilies gives rise to restricted transit equivalences with non trivial
(even infinite) quotient sets. We analyze them in terms of certain preserved
structures of differential topological nature carried by any branched
triangulations; in particular a pair of transverse foliations with determined
singular sets contained in V, including as particular cases the configuration
of the vertical and horizontal foliations of the square of an Abelian
differential on a Riemann surface.Comment: 22 pages, 11 figure
- …