Cutting polytopes and flag \(f\)-vectors (Q1971508)

The \({\mathbf c}{\mathbf d}\)-index of a polytope \(P\) encodes all the entries of the flag \(f\)-vector of \(P\). However, even standard operations on polytopes, such as the join and direct product, lead to complicated expressions for the resulting \({\mathbf c}{\mathbf d}\)-indices. Here, the authors show how to find the \({\mathbf c}{\mathbf d}\)-index of the polytope obtained by cutting off a face \(F\) of \(P\), by which is meant moving a hyperplane which supports \(P\) in \(F\) a little into \(P\). On the way, they find the \({\mathbf c}{\mathbf d}\)-index of the cell complex obtained by contracting \(F\) to a point.