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Generating Smooth Lattice Polytopes

  • Christian Haase
  • Benjamin Lorenz
  • Andreas Paffenholz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6327)

Abstract

A lattice polytope P is the convex hull of finitely many lattice points in ℤ d . It is smooth if each cone in the normal fan is unimodular. It has recently been shown that in fixed dimension the number of lattice equivalence classes of smooth lattice polytopes in dimension d with at most N lattice points is finite. We describe an algorithm to compute a representative in each equivalence class, and report on results in dimension 2 and 3 for N ≤ 12. Our algorithm is implemented as an extension to the software system polymake.

Keywords

lattice polytopes smooth polytopes classification polymake 

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References

  1. 1.
    Aardal, K., Weismantel, R., Wolsey, L.A.: Non-standard approaches to integer programming. Discrete Applied Mathematics 123(1-3), 5–74 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Batyrev, V.: On the classification of toric Fano 4-folds. Journal of Mathematical Sciences 94(1), 1021–1050 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Batyrev, V., Kreuzer, M.: Integral cohomology and mirror symmetry for Calabi-Yau 3-folds. In: Mirror symmetry. V. AMS/IP Stud. Adv. Math., vol. 38, pp. 255–270. Amer. Math. Soc., Providence (2006)Google Scholar
  4. 4.
    Beck, M., Chen, B., Fukshansky, L., Haase, C., Knutson, A., Reznick, B., Robins, S., Schürmann, A.: Problems from the Cottonwood Room. Contemporary Mathematics 374, 179–191 (2005)Google Scholar
  5. 5.
    Bogart, T., Haase, C., Hering, M., Lorenz, B., MacLagan, D., Nill, B., Paffenholz, A., Santos, F., Schenck, H.: Few smooth d-polytopes with n lattice points (May 2010) (in preparation)Google Scholar
  6. 6.
    Bruns, W., Gubeladze, J.: Polytopes, rings, and K-theory. Springer Monographs in Mathematics. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  7. 7.
    De Loera, J.A., Hemmecke, R., Yoshida, R., Tauzer, J.: lattE, http://www.math.ucdavis.edu/~latte/
  8. 8.
    Diaconis, P., Sturmfels, B.: Algebraic algorithms for sampling from conditional distributions. Annals of Statistics 26(1), 363–397 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gawrilow, E., Joswig, M.: polymake, http://www.opt.tu-darmstadt.de/polymake/
  10. 10.
    Gubeladze, J.: Convex normality of rational polytopes with long edges (December 2009), arxiv.org/abs/0912.1068Google Scholar
  11. 11.
    Haase, C., Hibi, T., MacLagan, D. (eds.): Mini-Workshop: Projective normality of smooth toric varieties, Oberwolfach reports, vol. 4 (2007)Google Scholar
  12. 12.
    Haase, C., Nill, B., Paffenholz, A., Santos, F.: Lattice points in Minkowski sums. Electronic Journal of Combinatorics 15 (2008)Google Scholar
  13. 13.
    Haase, C., Paffenholz, A.: Quadratic Gröbner bases for smooth 3 x 3 transportation polytopes. Journal of Algebraic Combinatorics 30(4) (2009)Google Scholar
  14. 14.
    Haase, C., Schicho, J.: Lattice polygons and the number 2i + 7. American Mathematical Monthly 116(2), 151–165 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Joswig, M., Müller, B., Paffenholz, A.: Polymake and lattice polytopes. In: DMTCS Proceedings of FPSAC (February 2009)Google Scholar
  16. 16.
    Kleinschmidt, P.: A classification of toric varieties with few generators. Aequationes Mathematicae 35(2-3), 254–266 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kreuzer, M., Skarke, H.: On the classification of reflexive polyhedra. Communications in Mathematical Physics 185(2), 495–508 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kreuzer, M., Skarke, H.: PALP: A package for analyzing lattice polytopes with applications to toric geometry (April 2002), http://hep.itp.tuwien.ac.at/~kreuzer/CY/CYpalp.html
  19. 19.
    Lorenz, B.: Generating smooth polytopes - extension for polymake, http://ehrhart.math.fu-berlin.de/people/benmuell/classification.html
  20. 20.
    Lorenz, B.: Classification of smooth lattice polytopes with few lattice points. Diploma thesis (2010), arxiv.org/abs/1001.0514Google Scholar
  21. 21.
    Lutz, F.: The manifold page (April 2010), http://www.math.tu-berlin.de/diskregeom/stellar/
  22. 22.
    McDuff, D.: Displacing lagrangian toric fibers via probes (April 2009), arxiv.org/abs/0904.1686Google Scholar
  23. 23.
    Nill, B., Paffenholz, A.: Examples of non-symmetric Kähler-Einstein toric Fano manifolds (May 2009), arxiv.org/abs/0905.2054Google Scholar
  24. 24.
    Øbro, M.: An algorithm for the classification of smooth Fano polytopes (April 2007), arxiv.org/abs/0704.0049Google Scholar
  25. 25.
    Oda, T.: Convex Bodies and Algebraic Geometry. In: An Introduction to the Theory of Toric Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Heidelberg (1987)Google Scholar
  26. 26.
    Oda, T.: Problems on Minkowski sums of convex lattice polytopes (December 2008), arxiv.org/abs/0812.1418Google Scholar
  27. 27.
    Ohsugi, H., Hibi, T.: Unimodular triangulations and coverings of configurations arising from root systems. J. Algebr. Comb. 14(3), 199–219 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Sturmfels, B.: Equations defining toric varieties. In: Kollár, J. (ed.) Algebraic geometry. Proc. Summer Research Institute, Santa Cruz, CA, USA, July 9-29 (1995); Proc. Symp. Pure Math., vol. 62, pp. 437–449. AMS, Providence (1997)Google Scholar
  29. 29.
    Włodarczyk, J.: Toroidal varieties and the weak factorization theorem. Inventiones Mathematicae 154(2), 223–331 (2003)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Christian Haase
    • 1
  • Benjamin Lorenz
    • 1
  • Andreas Paffenholz
    • 1
  1. 1.Freie Universität BerlinBerlinGermany

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