How to make a triangulation of S^3 polytopal
  We introduce a numerical isomorphism invariant p(T) for any triangulation T of S^3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T.   Specifically, if T is polytopal or shellable then p(T) is `small' in the sense that we obtain a linear upper bound for p(T) in the number n=n(T) of tetrahedra of T.   Conversely, if p(T) is `small' then T is `almost' polytopal, since we show how to transform T into a polytopal triangulation by O((p(T))^2) local subdivisions. The minimal number of local subdivisions needed to transform T into a polytopal triangulation is at least $\frac{p(T)}{3n}-n-2$.   Using our previous results [math.GT/0007032], we obtain a general upper bound for p(T) exponential in n^2. We prove here by explicit constructions that there is no general subexponential upper bound for p(T) in n. Thus, we obtain triangulations that are `very far' from being polytopal.   Our results yield a recognition algorithm for S^3 that is conceptually simpler, though somewhat slower, as the famous Rubinstein-Thompson algorithm. 
Simon A. King