  The bigraded Betti numbers b^{-i,2j}(P) of a simple polytope P are the
dimensions of the bigraded components of the Tor groups of the face ring k[P].
The numbers b^{-i,2j}(P) reflect the combinatorial structure of P as well as
the topology of the corresponding moment-angle manifold \mathcal Z_P, and
therefore they find numerous applications in combinatorial commutative algebra
and toric topology. Here we calculate some bigraded Betti numbers of the type
\beta^{-i,2(i+1)} for associahedra, and relate the calculation of the bigraded
Betti numbers for truncation polytopes to the topology of their moment-angle
manifolds. These two series of simple polytopes provide conjectural extrema for
the values of b^{-i,2j}(P) among all simple polytopes P with the fixed
dimension and number of vertices.
