Bigraded Betti Numbers of Certain Simple Polytopes

The bigraded Betti numbers $\beta^{-i,2j}(P)$ of a simple polytope $P$ are the dimensions of the bigraded components of the Tor groups of the face ring $\mathbf{k}[P]$. The numbers $\beta^{-i,2j}(P)$ reflect the combinatorial structure of $P$, as well as the topological structure of the corresponding moment-angle manifold $\mathcal Z_P$; thus, they find numerous applications in combinatorial commutative algebra and toric topology. We calculate certain bigraded Betti numbers of the type $\beta^{-i,2(i+1)}$ for associahedra and apply the calculation of bigraded Betti numbers for truncation polytopes to study the topology of their moment-angle manifolds. Presumably, for these two series of simple polytopes, the numbers $\beta^{-i,2j}(P)$ attain their minimum and maximum values among all simple polytopes $P$ of fixed dimension with a given number of facets.