Minimal face numbers for volume rigidity

Jack Southgate

Necessary and/or sufficient conditions for the generic rigidity of objects such as graphs, simplicial complexes in terms of their combinatorics are useful tools in rigidity theory. Here we present a necessary condition for the generic volume rigidity of a $d$-dimensional simplicial complex in $\mathbb{R}^d$ in terms of its $f$-vector and outline a proof that relies (almost entirely) on early undergraduate-level maths.