Skeletal rigidity of simplicial complexes, II

Abstract

This is the second part of a two-part paper, the first part of which appeared in an earlier issue of this journal. The notation and terminology follow those of the earlier part. The paper concerns a generalization of infinitesimal rigidity from a graph (or one-dimensional simplicial complex) embedded in d-space to a higher-dimensional simplicial complex, again embedded in d-space. This part begins with a section on coning, an important construction which preserves rigidity and stress. Then we investigate the connections with the g-theorem, which characterizes the possible f-vectors of simplicial polytopes. This connection, and the possibility of a combinatorial proof of the g-theorem which it provides, was the original motivation behind the entire paper. Then we give two additional versions of r-rigidity and r-stress, which are equivalent to the three versions already given in part I. We conclude with a discussion of avenues for further work.