  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.
