Before revision:  We also introduce two new complexes which we call the Alpha clique and Minibox complexes . We provide results on topological properties of these, as well as computational experiments which show that these can often be used to reduce the number of high-dimensional simplices included in Cech filtrations  and so speed up the computation of persistent homology .
After revision:  We also introduce two new complexes which we call the Alpha clique and Minibox complexes , and prove their equivalence with \v{C which show that these can often be used to reduce the number of high-dimensional simplices included in Cech filtrations and so speed up the computation of persistent homology .