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