@article {IOPORT.02091254,
    author       = {King, Simon A.},
    title        = {How to make a triangulation of $S^3$ polytopal.},
    year         = {2004},
    journal      = {Transactions of the American Mathematical Society},
    volume       = {356},
    number       = {11},
    issn         = {0002-9947},
    pages        = {4519-4542},
    publisher    = {American Mathematical Society, Providence, RI},
    doi          = {10.1090/S0002-9947-04-03465-8},
    abstract     = {Author's abstract: We introduce a numerical isomorphism invariant $p({\cal T})$ for any triangulation ${\cal T}$ of $S^3$. Although its definition is purely topological (inspired by the bridge number of knots), $p({\cal T})$ reflects the geometric properties of ${\cal T}$. Specifically, if ${\cal T}$ is polytopal or shellable, then $p({\cal T})$ is ``small'' in the sense that we obtain a linear upper bound for $p({\cal T})$ in the number $n=n ({\cal T})$ of tetrahedra of ${\cal T}$. Conversely, if $p({\cal T})$ is ``small'', then ${\cal T}$ is ``almost'' polytopal, since we show how to transform ${\cal T}$ into a polytopal triangulation by $O((p({\cal T}))^2)$ local subdivisions. The minimal number of local subdivisions needed to transform ${\cal T}$ into a polytopal triangulation is at least $\frac{p({\cal T})}{3n}-n-2$. Using our previous results [Geom. Topol. 5, 369--398 (2001; Zbl 1021.57005)], we obtain a general upper bound for $p({\cal T})$ exponential in $n^2$. We prove here by explicit constructions that there is no general subexponential upper bound for $p({\cal T})$ in $n$. Thus, we obtain triangulatious 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.},
    reviewer     = {Horst Martini (Chemnitz)},
    identifier   = {02091254},
}