How to make a triangulation of $S^3$ polytopal
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- by Simon A. King PDF
- Trans. Amer. Math. Soc. 356 (2004), 4519-4542 Request permission
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.References
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Additional Information
- Simon A. King
- Affiliation: Department of Mathematics, Darmstadt University of Technology, Schlossgartenstr. 7, 64289 Darmstadt, Germany
- Email: king@mathematik.tu-darmstadt.de
- Received by editor(s): May 28, 2002
- Received by editor(s) in revised form: July 1, 2003
- Published electronically: February 27, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 4519-4542
- MSC (2000): Primary 52B11, 57M25; Secondary 57M15, 05C10, 52B22
- DOI: https://doi.org/10.1090/S0002-9947-04-03465-8
- MathSciNet review: 2067132