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On the homeomorphism problem of trivalent 2-stratifolds with finite homology groups

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Abstract

Trivalent 2-stratifolds X with finite homology groups can be encoded as bicolored trivalent graphs \(\Gamma \). We show that the homeomorphism problem for such stratifolds can be solved with complexity O(||V||), where V is the set of vertices of \(\Gamma \).

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References

  1. Aho, A., Hopcroft, J., Ullman, J.: The Design and Analysis of Computer Algorithms, pp. 84–85. Publishing Co., Reading, MA (1974)

    Google Scholar 

  2. Bendich, P., Wang, B., Mukherjee, S.: Towards Stratification Learning through Homology Inference, ar**v:1008.3572v1 [math.GT] (2010)

  3. Bendich, P., Gasparovic, E., Tralie, C.J., Harer, J.: Scaffoldings and spines: organizing high-dimensional data using cover trees, local principal component analysis, and persistent homology. Res. Comput. Topol. 2018, 93–114 (2018)

    Article  MathSciNet  Google Scholar 

  4. Campbell, D.M., Radford, D.: Tree isomorphism algorithms: speed vs clarity. Math. Mag. 64(4), 252–261 (1991)

    Article  MathSciNet  Google Scholar 

  5. Carter, J.S.: Reidemeister/Roseman-type moves to embedded foams in 4-dimensional space, in "New Ideas in Low Dimensional Topology’’. Ser. Knots Everything 56, 1–30 (2015)

    Article  Google Scholar 

  6. Damiand, G., Peltier, S., Fuchs, L.: Computing Homology for Surfaces with Generalized Maps: Application to 3D Images. In: Advances in Visual Computing. ISVC. Lecture Notes in Computer Science, vol 4292. Springer, Berlin, Heidelberg (2006)

  7. Goldstein, R.E., McTavish, J., Moffatt, H.K., Pesci, A.I.: Boundary singularities produced by the motion of soap films (2014). https://doi.org/10.1073/pnas.1406385111

  8. Gómez-Larrañaga, J.C., González-Acuña, F., Heil, W.: Categorical group invariants of 3-manifolds. Manuscr. Math. 145, 433–448 (2014)

    Article  MathSciNet  Google Scholar 

  9. Gómez-Larrañaga, J.C., González-Acuña, F.: Wolfgang Heil, \(2\)-stratifolds, in “A Mathematical Tribute to José María Montesinos Amilibia", Universidad Complutense de Madrid, 395-405 (2016)

  10. Gómez-Larrañaga, J.C., González-Acuña, F., Heil, W..: Classification of Simply-connected Trivalent \(2\)-dimensional Stratifolds. Top. Proc. 52, 329–340 (2018)

    MathSciNet  Google Scholar 

  11. Gómez-Larrañaga, J.C., González-Acuña, F., Heil, W.: Models of simply-connected trivalent 2-dimensional stratifolds. Boletín de la Sociedad Matemática Mexicana 26(3), 1301–1312 (2020)

    Article  MathSciNet  Google Scholar 

  12. Khovanov, M.: sl(3) link homology. Algebr. and Geom. Topol. 4, 1045–1081 (2004)

    Article  MathSciNet  Google Scholar 

  13. Hernández-Ketchul, M., Rodriguez-Viorato, J.: String representation of trivalent 2-stratifolds with trivial fundamental group. Bol. Soc. Mat. Mex. 29, 22 (2023)

    Article  MathSciNet  Google Scholar 

  14. Lum, P., Singh, G., Carlsson, J., Lehman, A., Ishkhanov, T., Vejdemo-Johansson, M., Alagappan, M., Carlsson, G.: Extracting insights from the shape of complex data using topology. Nat. Sci. Rep. 3, 12–36 (2013)

    Google Scholar 

  15. Martin, S., Thompson, A., Coutsias, E.A., Watson, J.P.: Topology of cyclo-octane energy landscape. J. Chem. Phys. 132, 234115 (2010)

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  16. Matsuzaki, S., Ozawa, M.: Genera and minors of multibranched surfaces. Topol. Appl. 230, 621–638 (2017)

    Article  MathSciNet  Google Scholar 

  17. Matveev, S.: Algorithmic Topology and Classification of 3-Manifolds, Algorithms and Computation in Mathematics AACIM, vol. 9. Springer, Berlin (2003)

    Google Scholar 

  18. Piergallini, R.: Standard moves for standard polyhedra and spines, Third National Conference on Topology (Italian) (Trieste, 1986). Rend. Circ. Mat. Palermo (2) Suppl. No. 18 (1988), 391-414

  19. Neumann, W.: A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Trans. Amer. Math. Soc. 268, 299–344 (1981)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Centro de Investigación en Matemáticas, Mérida, México

    J. C. Gómez-Larrañaga

  2. Instituto de Matemáticas, UNAM, 62210, Cuernavaca, Morelos, México

    F. González-Acuña

  3. Centro de Investigación en Matemáticas, A.P. 402, Guanajuato, 36000, Gto, México

    F. González-Acuña

  4. Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

    Wolfgang Heil

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  1. J. C. Gómez-Larrañaga
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  2. F. González-Acuña
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  3. Wolfgang Heil
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Correspondence to Wolfgang Heil.

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Gómez-Larrañaga, J.C., González-Acuña, F. & Heil, W. On the homeomorphism problem of trivalent 2-stratifolds with finite homology groups. Bol. Soc. Mat. Mex. 30, 17 (2024). https://doi.org/10.1007/s40590-023-00588-y

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  • DOI: https://doi.org/10.1007/s40590-023-00588-y

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