How to make a triangulation of $S^3$ polytopal

We introduce a numerical isomorphism invariant $p(\mathcal{T})$ for any triangulation $\mathcal{T}$ of $S^3$. Although its definition is purely topological (inspired by the bridge number of knots), $p(\mathcal{T})$ reflects the geometric properties of $\mathcal{T}$. Specifically, if $\mathcal{T}$ is polytopal or shellable, then $p(\mathcal{T})$is ``small'' in the sense that we obtain a linear upper bound for $p(\mathcal{T})$ in the number $n=n(\mathcal{T})$ of tetrahedra of $\mathcal{T}$. Conversely, if $p(\mathcal{T})$ is ``small'', then $\mathcal{T}$is ``almost'' polytopal, since we show how to transform $\mathcal{T}$ into a polytopal triangulation by $O((p(\mathcal{T}))^2)$ local subdivisions. The minimal number of local subdivisions needed to transform $\mathcal{T}$ into a polytopal triangulation is at least $\frac{p(\mathcal{T})}{3n}-n-2$.

Using our previous results [The size of triangulations supporting a given link, Geometry & Topology 5 (2001), 369-398], we obtain a general upper bound for $p(\mathcal{T})$ exponential in $n^2$. We prove here by explicit constructions that there is no general subexponential upper bound for $p(\mathcal{T})$ 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, although somewhat slower, than the famous Rubinstein-Thompson algorithm.