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

Abstract:

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.