Bigraded Betti numbers of some simple polytopes

TL;DR

This paper computes specific bigraded Betti numbers for associahedra and truncation polytopes, revealing their extremal properties and connections to the topology of associated moment-angle manifolds.

Contribution

It provides new calculations of bigraded Betti numbers for certain simple polytopes and links these to topological and combinatorial extremal properties.

Findings

Calculated Betti numbers for associahedra and truncation polytopes.

Identified these polytopes as conjectural extrema for Betti number values.

Connected Betti number calculations to the topology of moment-angle manifolds.

Abstract

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.