Francisco Santos: Constructing the multi-associahedron: some ideas that did not work

Abstract:
A set of $k+1$ diagonals of a convex $n$-gon is called a
$k+1$-crossing if every pair of them cross. A maximal family of diagonals
of the $n$-gon not containing a $k+1$-crossing is a $k$-triangulation. The
family of all $k$-triangulations of the $n$-gon is a shellable sphere of
dimension $k(n-2k-1)$, and conjectured to be polytopal. Such a polytope,
in case it exists, is called the multi-associahedron.
In this talk I will review several attempts of constructing the
multi-associahedron, based on trying to adapt to the "multi" case the
ideas that worked in the "single" case. None of them (is known to) work,
but they are interesting anyway. In particular, $k$-triangulations happen
to be $(2k,{k+1 \choose 2})$-tight graphs, which is a necessary (and
typically sufficient) condition fo a graph to be generically rigid and
stress-free (that is to say, generically $2k$-isostatic) when embedded in
dimension $2k$. This raises the conjecture that $k$-triangulations are
indeed $2k$-isostatic and opens the door to constructing the associahedron
as a ``polytope of expansive motions'' following the ideas of Rote, Santos
and Streinu.
This is joint work with Vincent Pilaud.