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Generalised cone complexes and tropical moduli in polymake

Published: 24 July 2023 Publication History

Abstract

We investigate geometric embeddings among several classes of generalised cone complexes and algorithms, e.g., to compute their homology. Interesting cases arise from moduli spaces of tropical curves. Specifically, via an explicit computation, we show that the tropical honeycomb curves form a contractible sub-locus in the moduli of all tropical K4-curves.

References

[1]
Dan Abramovich, Lucia Caporaso, and Sam Payne. 2015. The tropicalization of the moduli space of curves. Ann. Sci. Éc. Norm. Supér. (4) 48, 4 (2015), 765–809. https://doi.org/10.24033/asens.2258
[2]
Daniel Allcock, Daniel Corey, and Sam Payne. 2022. Tropical moduli spaces as symmetric Δ -complexes. Bull. Lond. Math. Soc. 54, 1 (2022), 193–205. https://doi.org/10.1112/blms.12570
[3]
Bruno Benedetti and Frank H. Lutz. 2014. Random discrete Morse theory and a new library of triangulations. Exp. Math. 23, 1 (2014), 66–94. https://doi.org/10.1080/10586458.2013.865281
[4]
Anders Björner. 1995. Topological methods. In Handbook of combinatorics, Vol. 1, 2. Elsevier Sci. B. V., Amsterdam, 1819–1872.
[5]
Glen E. Bredon. 1972. Introduction to compact transformation groups. Academic Press, New York-London.
[6]
Sarah Brodsky, Michael Joswig, Ralph Morrison, and Bernd Sturmfels. 2015. Moduli of tropical plane curves. Res. Math. Sci. 2, 4 (2015). https://doi.org/10.1186/s40687-014-0018-1
[7]
Melody Chan. 2012. Combinatorics of the tropical Torelli map. Algebra Number Theory 6, 6 (2012), 1133–1169. https://doi.org/10.2140/ant.2012.6.1133
[8]
Melody Chan. 2017. Lectures on tropical curves and their moduli spaces. In Moduli of curves. Lect. Notes Unione Mat. Ital., Vol. 21. Springer, Cham, 1–26.
[9]
Melody Chan, Søren Galatius, and Sam Payne. 2021. Tropical curves, graph homology, and top weight cohomology of . J. Amer. Math. Soc. 34, 2 (2021), 565–594. https://doi.org/10.1090/jams/965
[10]
Melody Chan and Pakawut Jiradilok. 2017. Theta characteristics of tropical K4-curves. In Combinatorial algebraic geometry. Fields Inst. Commun., Vol. 80. Fields Inst. Res. Math. Sci., Toronto, ON, 65–86.
[11]
Manoj K. Chari. 2000. On discrete Morse functions and combinatorial decompositions. Discrete Math. 217, 1-3 (2000), 101–113. https://doi.org/10.1016/S0012-365X(99)00258-7
[12]
Robin Forman. 1998. Morse theory for cell complexes. Adv. Math. 134, 1 (1998), 90–145. https://doi.org/10.1006/aima.1997.1650
[13]
Ewgenij Gawrilow and Michael Joswig. 2000. polymake: a framework for analyzing convex polytopes. In Polytopes—combinatorics and computation (Oberwolfach, 1997). DMV Sem., Vol. 29. Birkhäuser, Basel, 43–73.
[14]
Costas S. Iliopoulos. 1989. Worst-case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix. SIAM J. Comput. 18, 4 (1989), 658–669. https://doi.org/10.1137/0218045
[15]
Michael Joswig. 2021. Essentials of tropical combinatorics. Graduate Studies in Mathematics, Vol. 219. American Mathematical Society, Providence, RI.
[16]
Michael Joswig, Davide Lofano, Frank H. Lutz, and Mimi Tsuruga. 2022. Frontiers of sphere recognition in practice. J. Appl. Comput. Topol. 6, 4 (2022), 503–527. https://doi.org/10.1007/s41468-022-00092-8
[17]
Michael Joswig and Marc E. Pfetsch. 2006. Computing optimal Morse matchings. SIAM J. Discrete Math. 20, 1 (2006), 11–25. https://doi.org/10.1137/S0895480104445885
[18]
Michael Joswig and Ayush Kumar Tewari. 2021. Forbidden patterns in tropical plane curves. Beitr. Algebra Geom. 62, 1 (2021), 65–81. https://doi.org/10.1007/s13366-020-00523-6
[19]
Michael Joswig and Thorsten Theobald. 2013. Polyhedral and Algebraic Methods in Computational Geometry. Springer.
[20]
Marek Kaluba, Benjamin Lorenz, and Sascha Timme. 2020. Polymake.jl: A New Interface to polymake. In Mathematical Software - ICMS 2020 - 7th International Conference, Braunschweig, Germany, July 13-16, 2020, Proceedings(Lecture Notes in Computer Science, Vol. 12097), Anna Maria Bigatti, Jacques Carette, James H. Davenport, Michael Joswig, and Timo de Wolff (Eds.). Springer, 377–385. https://doi.org/10.1007/978-3-030-52200-1_37
[21]
Ravindran Kannan and Achim Bachem. 1979. Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8, 4 (1979), 499–507. https://doi.org/10.1137/0208040
[22]
Eduard Looijenga. 1993. Cohomology of M3 and . In Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991). Contemp. Math., Vol. 150. Amer. Math. Soc., Providence, RI, 205–228. https://doi.org/10.1090/conm/150/01292
[23]
Diane Maclagan and Bernd Sturmfels. 2015. Introduction to tropical geometry. Graduate Studies in Mathematics, Vol. 161. American Mathematical Society, Providence, RI.
[24]
James R. Munkres. 1984. Elements of Algebraic Topology. Addison–Wesley Publishing Company, Menlo Park, CA.
[25]
Martin Tancer. 2016. Recognition of collapsible complexes is NP-complete. Discrete Comput. Geom. 55, 1 (2016), 21–38. https://doi.org/10.1007/s00454-015-9747-1
[26]
The OSCAR Team. 2023. OSCAR– Open Source Computer Algebra Research system, version 0.12.0. https://oscar.computeralgebra.de.
[27]
Claudia He Yun. 2022. Discrete Morse theory for symmetric Delta-complexes. Preprint arXiv:2209.01070.
[28]
Günter M. Ziegler. 1995. Lectures on polytopes. Graduate Texts in Mathematics, Vol. 152. Springer-Verlag, New York.

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cover image ACM Other conferences
ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation
July 2023
567 pages
ISBN:9798400700392
DOI:10.1145/3597066
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Association for Computing Machinery

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Publication History

Published: 24 July 2023

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Author Tags

  1. embeddings
  2. homology of moduli spaces
  3. tropical curves

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