FU Berlin

Associahedra Seminar

December 17, 2012

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.