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Nerve Complexes of Circular Arcs

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 \(O(n\log n)\).

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Notes

  1. 1.

    Attali and Lieutier [4] refer to the ambient Čech complex as a restricted Čech complex.

References

  1. 1.

    Adamaszek, M.: Clique complexes and graph powers. Isr. J. Math. 196(1), 295–319 (2013)

  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)

  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)

  6. 6.

    Babenko, A.G.: An extremal problem for polynomials. Math. Notes 35(3), 181–186 (1984)

  7. 7.

    Babson, E., Kozlov, D.N.: Complexes of graph homomorphisms. Isr. J. Math. 152(1), 285–312 (2006)

  8. 8.

    Bagchi, B., Datta, B.: Minimal triangulations of sphere bundles over the circle. J. Comb. Theory, Ser. A 115(5), 737–752 (2008)

  9. 9.

    Barmak, J.A.: On Quillen’s Theorem A for posets. J. Comb. Theory, Ser. A 118(8), 2445–2453 (2011)

  10. 10.

    Barmak, J.A., Minian, E.G.: Strong homotopy types, nerves and collapses. Discrete Comput. Geom. 47(2), 301–328 (2012)

  11. 11.

    Björner, A.: Topological Methods. Handbook of Combinatorics, vol. 2. Elsevier, Amsterdam (1995)

  12. 12.

    Borsuk, K.: On the imbedding of systems of compacta in simplicial complexes. Fundam. Math. 35(1), 217–234 (1948)

  13. 13.

    Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)

  14. 14.

    Chazal, F., de Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geom. Dedicata 173(1), 193–214 (2013)

  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)

  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)

  17. 17.

    Edelsbrunner, H., Harer, J.L.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)

  18. 18.

    Gale, D.: Neighborly and cyclic polytopes. Proc. Symp. Pure Math. 7, 225–232 (1963)

  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)

  20. 20.

    Golumbic, M.C., Hammer, P.L.: Stability in circular arc graphs. J. Algorithms 9(3), 314–320 (1988)

  21. 21.

    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

  22. 22.

    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)

  23. 23.

    Kozlov, D.N.: Combinatorial Algebraic Topology. Algorithms and Computation in Mathematics, vol. 21. Springer, Berlin (2008)

  24. 24.

    Kozma, G., Oravecz, F.: On the gaps between zeros of trigonometric polynomials. Real Anal. Exch. 28(2), 447–454 (2002)

  25. 25.

    Kühnel, W.: Higherdimensional analogues of Császár’s torus. Result. Math. 9, 95–106 (1986)

  26. 26.

    Kühnel, W., Lassmann, G.: Permuted difference cycles and triangulated sphere bundles. Discrete Math. 162(1–3), 215–227 (1996)

  27. 27.

    Latschev, J.: Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold. Arch. Math. 77(6), 522–528 (2001)

  28. 28.

    Lovász, L.: Kneser’s conjecture, chromatic number, and homotopy. J. Comb. Theory, Ser. A 25(3), 319–324 (1978)

  29. 29.

    Matoušek, J.: LC reductions yield isomorphic simplicial complexes. Contrib. Discrete Math. 3(2), 37–39 (2008)

  30. 30.

    Montgomery, H.L., Ulrike, M.A.: Biased trigonometric polynomials. Am. Math. Mon. 114(9), 804–809 (2007)

  31. 31.

    Previte-Johnson, C.: The \(D\)-Neighborhood Complex of a Graph. PhD thesis, Colorado State University, Fort Collins (2014)

  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)

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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.

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Correspondence to Florian Frick.

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Editor in Charge: Günter M. Ziegler

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Adamaszek, M., Adams, H., Frick, F. et al. Nerve Complexes of Circular Arcs. Discrete Comput Geom 56, 251–273 (2016). https://doi.org/10.1007/s00454-016-9803-5

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Keywords

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

Mathematics Subject Classification

  • 05E45
  • 52B15
  • 68R05