In this joint paper that I should be working on, we make significant use of a certain generalization of a triangulated disk. Many of our important examples are triangulated disks, but we are also interested in certain simplicial complexes that are singular disks, or even more generally singular disks tiled by polygons. It is easy to describe the disks of interest to us as simplicial complexes which are contractible, compact subsets of the plane. The embedding in the plane only matters at all up to isotopy, and it also does not matter all that much; the most important condition is that the complex is contractible and at most two triangles meet at every edge. For instance, you can have two triangulated disks that meet at a vertex, trees, "barbells", etc.
One could more generally look at those topological spaces in the plane that are an intersection of nested, closed disks, or maybe those that are locally connected. For instance, the Mandelbrot set is one. We do not need them for what we are doing if they are not simplicial complexes, but this is an interesting class of topological spaces that should have a good name.
The following names have been proposed:
- Van Kampen diagram - standard but ugly; very close to what we use although our edges are not labeled by group elements.
- Contractible plane continuum - descriptive but clumsy.
- Diskoid - a good name, by analogy with a dendroid which is the 1-dimensional case, but it doesn't seem to be standard.
- Cactus - also a good name, by analogy with trees in graph theory, but it seems non-standard.
- Singular disk - livable but not specific enough.
The reason that I want a short name is that the object X itself is used to make a moduli space or an algebraic variety. The moduli space is easy to describe: Assign fixed lengths to the edges of X and look at its rigid embeddings into a metric space. So we wouldn't want to say "contractible plane continuum variety". Also, for no particular reason I've been thinking of the triangulation as an extra decoration and instead name the underlying topological space.
I can certainly think of livable names, but the general idea appears in several places in mathematics and I would prefer a good name. Could someone with a good sense of the literature argue for a particular term, not necessarily one of the ones listed above? Or any useful opinion would be welcome.