Advertisement

Discrete & Computational Geometry

, Volume 56, Issue 2, pp 251–273 | Cite as

Nerve Complexes of Circular Arcs

  • Michał Adamaszek
  • Henry Adams
  • Florian Frick
  • Chris Peterson
  • Corrine Previte-Johnson
Article

Abstract

We show that the nerve and clique complexes of n arcs in the circle are homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time \(O(n\log n)\). For the particular case of the nerve complex of evenly-spaced arcs of the same length, we determine explicit homology bases and we relate the complex to a cyclic polytope with n vertices. We give three applications of our knowledge of the homotopy types of nerve complexes of circular arcs. First, we show that the Lovász bound on the chromatic number of a circular complete graph is either sharp or off by one. Second, we use the connection to cyclic polytopes to give a novel topological proof of a known upper bound on the distance between successive roots of a homogeneous trigonometric polynomial. Third, we show that the Vietoris–Rips or ambient Čech simplicial complex of n points in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension, and furthermore this homotopy type can be computed in time \(O(n\log n)\).

Keywords

Nerve complex Čech complex Vietoris–Rips complex Circular arc Cyclic polytope 

Mathematics Subject Classification

05E45 52B15 68R05 

Notes

Acknowledgments

We would like to thank Anton Dochtermann for encouraging us to consider the connection to the Lovász bound in Sect. 6, and we would like to thank Arnau Padrol and Yuliy Baryshnikov for helpful conversations about cyclic polytopes. We are grateful to the referees for suggestions regarding the paper, and in particular for bringing [30] to our attention. Research of MA was carried out while at the Max Planck Institut für Informatik, Saarbrücken, Germany. Research of HA was supported by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation. FF was supported by the German Science Foundation DFG via the Berlin Mathematical School.

References

  1. 1.
    Adamaszek, M.: Clique complexes and graph powers. Isr. J. Math. 196(1), 295–319 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adamaszek, M., Adams, H.: The Vietoris–Rips complex of the circle. Preprint, arXiv:1503.03669
  3. 3.
    Adamaszek, M., Adams, H., Motta, F.: Random cyclic dynamical systems. Preprint, arXiv:1511.07832
  4. 4.
    Attali, D., Lieutier, A.: Geometry driven collapses for converting a Čech complex into a triangulation of a shape. Discrete Comput. Geom. 54(4), 798–825 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Attali, D., Lieutier, A., Salinas, D.: Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes. Comput. Geom. 46(4), 448–465 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Babenko, A.G.: An extremal problem for polynomials. Math. Notes 35(3), 181–186 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Babson, E., Kozlov, D.N.: Complexes of graph homomorphisms. Isr. J. Math. 152(1), 285–312 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bagchi, B., Datta, B.: Minimal triangulations of sphere bundles over the circle. J. Comb. Theory, Ser. A 115(5), 737–752 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Barmak, J.A.: On Quillen’s Theorem A for posets. J. Comb. Theory, Ser. A 118(8), 2445–2453 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Barmak, J.A., Minian, E.G.: Strong homotopy types, nerves and collapses. Discrete Comput. Geom. 47(2), 301–328 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Björner, A.: Topological Methods. Handbook of Combinatorics, vol. 2. Elsevier, Amsterdam (1995)Google Scholar
  12. 12.
    Borsuk, K.: On the imbedding of systems of compacta in simplicial complexes. Fundam. Math. 35(1), 217–234 (1948)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chazal, F., de Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geom. Dedicata 173(1), 193–214 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chazal, F., Oudot, S.: Towards persistence-based reconstruction in Euclidean spaces. In: Proceedings of the 24th Annual Symposium on Computational Geometry, pp. 232–241. ACM, New York (2008)Google Scholar
  16. 16.
    Colin de Verdière, É., Ginot, G., Goaoc, X.: Multinerves and Helly numbers of acyclic families. In: Proceedings of the 28th Annual Symposium on Computational Geometry, pp. 209–218. ACM, New York (2012)Google Scholar
  17. 17.
    Edelsbrunner, H., Harer, J.L.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  18. 18.
    Gale, D.: Neighborly and cyclic polytopes. Proc. Symp. Pure Math. 7, 225–232 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gilbert, A.D., Smyth, C.J.: Zero-mean cosine polynomials which are non-negative for as long as possible. J. Lond. Math. Soc. 62(2), 489–504 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Golumbic, M.C., Hammer, P.L.: Stability in circular arc graphs. J. Algorithms 9(3), 314–320 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  22. 22.
    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)CrossRefzbMATHGoogle Scholar
  23. 23.
    Kozlov, D.N.: Combinatorial Algebraic Topology. Algorithms and Computation in Mathematics, vol. 21. Springer, Berlin (2008)zbMATHGoogle Scholar
  24. 24.
    Kozma, G., Oravecz, F.: On the gaps between zeros of trigonometric polynomials. Real Anal. Exch. 28(2), 447–454 (2002)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kühnel, W.: Higherdimensional analogues of Császár’s torus. Result. Math. 9, 95–106 (1986)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kühnel, W., Lassmann, G.: Permuted difference cycles and triangulated sphere bundles. Discrete Math. 162(1–3), 215–227 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Latschev, J.: Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold. Arch. Math. 77(6), 522–528 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lovász, L.: Kneser’s conjecture, chromatic number, and homotopy. J. Comb. Theory, Ser. A 25(3), 319–324 (1978)CrossRefzbMATHGoogle Scholar
  29. 29.
    Matoušek, J.: LC reductions yield isomorphic simplicial complexes. Contrib. Discrete Math. 3(2), 37–39 (2008)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Montgomery, H.L., Ulrike, M.A.: Biased trigonometric polynomials. Am. Math. Mon. 114(9), 804–809 (2007)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Previte-Johnson, C.: The \(D\)-Neighborhood Complex of a Graph. PhD thesis, Colorado State University, Fort Collins (2014)Google Scholar
  32. 32.
    Taylan, D.: Matching trees for simplicial complexes and homotopy type of devoid complexes of graphs. Order (2015). doi: 10.1007/s11083-015-9379-3
  33. 33.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, Berlin (1995)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Michał Adamaszek
    • 1
  • Henry Adams
    • 2
  • Florian Frick
    • 3
  • Chris Peterson
    • 2
  • Corrine Previte-Johnson
    • 4
  1. 1.Department of MathematicsUniversity of CopenhagenCopenhagenDenmark
  2. 2.Department of MathematicsColorado State UniversityFort CollinsUSA
  3. 3.Department of MathematicsCornell UniversityIthacaUSA
  4. 4.Department of MathematicsCalifornia State UniversitySan BernardinoUSA

Personalised recommendations