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Triangulations of cyclic polytopes and higher Bruhat orders

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  26 February 2010

Jörg Rambau
Jörg Rambau
Affiliation:
Konrad-Zuse-Zentrum für Informationstechnik, Takustr. 7, D-14195 Berlin, Germany. E-mail: [email protected]
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Abstract

Recently Edelman and Reiner suggested two poset structures, (n, d) and (n, d) on the set of all triangulations of the cyclic d-polytope C(n, d) with n vertices. Both posets are generalizations of the well-studied Tamari lattice. While (n, d) is bounded by definition, the same is not obvious for (n, d). In the paper by Edelman and Reiner the bounds of (n, d) were also confirmed for (n, d) whenever d≤5, leaving the general case as a conjecture.

MSC classification

Secondary: 52B70: Polyhedral manifolds
Type
Research Article
Information
Mathematika , Volume 44 , Issue 1 , June 1997 , pp. 162 - 194
Copyright
Copyright © University College London 1997

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References

1.Bayer, M. M.. Equidecomposable and weakly neighborly polytopes. Israel Journal of Mathematics, 81 (1993), 301–320.Google Scholar
2.Billera, L. J., Filliman, P. and Sturmfels, B., Constructions and complexity of secondary polytopes. Advances in Mathematics, 83 (1990), 155–179.Google Scholar
3.Billera, L. J., Gelfand, I. M. and Sturmfels, B.. Duality and minors of secondary polyhedra. Journal of Combinatorial Theory, Series B, 57 (1993), 258–268.Google Scholar
4.Billera, L. J., Kapranov, M. M. and Sturmfels, B.. Cellular strings on polytopes. Proceedings of the American Mathematical Society, 122 (1994), 549–555.Google Scholar
5.Björner, A., Vergnas, M. Las, Sturmfels, B., White, N. and Ziegler, G. M.. Oriented Matroids, Encyclopedia of Mathematics, Vol. 46 (Cambridge University Press, Cambridge, 1993)Google Scholar
6.Loera, J. de. Triangulations of Polytopes and Computational Algebra, Ph.D. thesis (Cornell University, 1995).Google Scholar
7.Edelman, P. H. and Reiner, V.. The higher Stasheff Tamari posets. Mathematika, 43 (1996), 127–154.Google Scholar
8.Edelsbrunner, H.. Algorithms in Geometry and Topology, Course script 497 (University of Illinois, Urbana-Champaign, Spring 1993).Google Scholar
9.Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V.. Discriminants of polynomials in several variables and triangulations of Newton polyhedra. Leningrad Mathematical Journal, 2 (1991), 449–505.Google Scholar
10.Grünbaum, B.. Convex Polytopes (Interscience, London, 1967).Google Scholar
11.Joe, B.. Three dimensional triangulations from local transformations. SIAM Journal of Scientific Statistical Computation, 10 (1989), 718–741.CrossRefGoogle Scholar
12.Kapranov, M. M. and Voevodsky, V. A.. Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results). Cahiers de Topologie et Géométrie différentielle catégoriques, 32 (1991), 11–27.Google Scholar
13.Lee, C.. Triangulations of polytopes. CRC Handjournal of Discrete and Computational Geometry (Goodman, J. E. and O'Rourke, J., eds.), (CRC-Press, 1997).Google Scholar
14.Loera, J. A. de, Hosten, S., Santos, F. and Sturmfels, B.. The polytope of all triangulations of a point configuration. Documenta Mathematika, (1996), 103–119.CrossRefGoogle Scholar
15.Manin, Yu. I. and Schechtman, V. V.. Arrangements of hyperplanes, higher braid groups and higher Bruhat orders. Advanced Studies in Pure Mathematics, 17 (1989), 289–308.Google Scholar
16.Rambau, J. and Ziegler, G. M.. Projections of polytopes and the Generalized Baues Conjecture. Discrete & Computational Geometry, 16.Google Scholar
17.Ziegler, G. M.. Higher Bruhat orders and cyclic hyperplane arrangements. Topology 32 (1993), 259–279.Google Scholar
18.Ziegler, G. M.. Shelling polyhedral 3-balls and 4-polytopes. To appear in Discrete & Computational Geometry.Google Scholar