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Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect

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Abstract

While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order.

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Notes

  1. Our notation is intentionally ambiguous with respect to smooth or topological genus, because all our arguments will apply equally well in both categories.

  2. Conway called them algebraic links, but this denomination is now more used for the links that come from algebraic curves in \(\mathbb {C}^2\).

References

  1. The knot atlas: \(8_{10}\). http://katlas.org/wiki/8_10. Accessed: 2024-13-03

  2. Agol, I., Hass, J., Thurston, W.: The computational complexity of knot genus and spanning area. Trans. Am. Math. Soc. 358(9), 3821–3850 (2006)

    Article  MathSciNet  Google Scholar 

  3. Baader, S.: Positive braids of maximal signature. Enseign. Math. (2) 59(3), 351–358 (2014)

    Article  MathSciNet  Google Scholar 

  4. Baader, S., Dehornoy, P.: Minor theory for surfaces and divides of maximal signature. arXiv preprint arXiv:1211.7348 (2012)

  5. Baader, S., Dehornoy, P., Liechti, L.: Minor theory for quasipositive surfaces. In: Athanase Papadopoulos, editor, Essays in geometry, dedicated to Norbert A’Campo, pages 351–358. EMS Pres (2023)

  6. Baader, S., Feller, P., Lewark, L., Liechti, L.: On the topological 4-genus of torus knots. Trans. Am. Math. Soc. 370(4), 2639–2656 (2018)

    Article  MathSciNet  Google Scholar 

  7. Baroni, F.: Classification of genus-two surfaces in \(\mathbb{S}^{3}\). arXiv preprint arXiv:2309.05387 (2023)

  8. Bastl, S., Burke, R., Chatterjee, R., Dey, S., Durst, A., Friedl, S., Galvin, D., Rivas, A.G., Hirsch, T., Hobohm, C., Hsueh, C.-S., Kegel, M., Kern, F., Lee, S.M.S., Löh, C., Manikandan, N., Mousseau, L., Munser, L., Pencovitch, M., Perras, P., Powell, M., Quintanilha, J.P., Schambeck, L., Suchodoll, D., Tancer, M., Thiele, A., Truöl, P., Uschold, M., Veselá, S., Weiß, M., von Wunsch-Rolshoven, M.: Algorithms in 4-manifold topology (2024). arXiv:2411.08775

  9. Bonahon, F., Siebenmann, L.C.: New geometric splittings of classical knots and the classification and symmetries of arborescent knots. Preprint available on the authors’ webpage (1979)

  10. Burde, G., Zieschang, H.: Knots. Walter de gruyter (2002)

  11. Burton, B.A., Budney, R., Pettersson, W., et al.: Regina: Software for low-dimensional topology. http://regina-normal.github.io/ (1999)–(2023)

  12. Cygan, M., Fomin, F.V., Kowalik, Ł, Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized algorithms, vol. 4. Springer, Berlin (2015)

    Book  Google Scholar 

  13. de Mesmay, A., Rieck, Y., Sedgwick, E., Tancer, M.: The unbearable hardness of unknotting. Adv. Math. 381, 107648 (2021)

    Article  MathSciNet  Google Scholar 

  14. Diestel, R.: Graph theory. Number 173 in Graduate texts in mathematics, 5th edn. Springer, New York (2016)

    Google Scholar 

  15. Fox, R.: Some problems in knot theory. In: Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute), pages 168–176, Englewood Cliffs, N.J. Prentice-Hall (1961)

  16. Freedman, M.H.: The topology of four-dimensional manifolds. Journal of Differential Geometry 17(3), 357–453 (1982)

    Article  MathSciNet  Google Scholar 

  17. Gabai, D.: The murasugi sum is a natural geometric operation. Contemp. Math. 20, 131–143 (1983)

    Article  MathSciNet  Google Scholar 

  18. Gabai, D.: Genera of the Arborescent Links. Memoirs of the American Mathematical Society. American Mathematical Society (1986)

  19. Giroux, E., Goodman, N.: On the stable equivalence of open books in three-manifolds. Geometry & Topology 10(1), 97–114 (2006)

    Article  MathSciNet  Google Scholar 

  20. McA, C., Gordon, Luecke, J.S.: Knots are determined by their complements. J. Am. Math. Soc., 2, 371–415 (1989)

  21. Haken, W.: Theorie der normalflächen. Acta Math. 105(3), 245–375 (1961)

    Article  MathSciNet  Google Scholar 

  22. Hass, J., Lagarias, J.C., Pippenger, N.: The computational complexity of knot and link problems. Journal of the ACM (JACM) 46(2), 185–211 (1999)

    Article  MathSciNet  Google Scholar 

  23. Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge; New York (2002)

    Google Scholar 

  24. Kronheimer, P.B., Mrowka, T.S.: The genus of embedded surfaces in the projective plane. Mathematical Research Letters 1(6), 797–808 (1994)

    Article  MathSciNet  Google Scholar 

  25. Kruskal, J.B.: Well-quasi-ordering, the tree theorem, and vazsonyi’s conjecture. Trans. Am. Math. Soc. 95, 210–225 (1960)

    MathSciNet  Google Scholar 

  26. Kuperberg, G.: Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization. Pac. J. Math. 301(1), 189–241 (2019)

    Article  MathSciNet  Google Scholar 

  27. Lackenby, M.: Elementary Knot Theory. In: Lectures on Geometry. Oxford University Press (01 2017)

  28. Lackenby, M.: Some conditionally hard problems on links and 3-manifolds. Discrete & Computational Geometry 58, 580–595 (2017)

    Article  MathSciNet  Google Scholar 

  29. Lackenby, M.: Algorithms in 3-manifold theory. Surveys in Differential Geometry (2020)

  30. Lackenby, M.: The efficient certification of knottedness and thurston norm. Adv. Math. 387, 107796 (2021)

    Article  MathSciNet  Google Scholar 

  31. Liechti, L.: On the genus defect of positive braid knots. Algebraic & Geometric Topology 20(1), 403–428 (2020)

    Article  MathSciNet  Google Scholar 

  32. Markov, A.A.: The insolubility of the problem of homeomorphy. In: Doklady Akademii Nauk, volume 121, pages 218–220. Russian Academy of Sciences (1958)

  33. Matoušek, J., Tancer, M., Wagner, U.: Hardness of embedding simplicial complexes in \(\mathbb{r} ^d\). J. Eur. Math. Soc. 13(2), 259–295 (2010)

    Article  Google Scholar 

  34. Matveev, S.V.: Algorithmic topology and classification of 3-manifolds. Springer, Berlin (2007)

    Google Scholar 

  35. Misev, F.: Cutting arcs for torus links and trees. Bull. Soc. Math. France 145, 575–602 (2014)

    Article  MathSciNet  Google Scholar 

  36. Misev, F.: On the plumbing structure of fibre surfaces. PhD thesis, Universität Bern (2016)

  37. Misev, F.: Hopf bands in arborescent hopf plumbings. Osaka Journal of Mathematics 56(2), 375–389 (2019)

    MathSciNet  Google Scholar 

  38. Murasugi, K.: On a certain subgroup of the group of an alternating link. Am. J. Math. 85(4), 544–550 (1963)

    Article  MathSciNet  Google Scholar 

  39. Nash-Williams, C.S.J.A.: On well-quasi-ordering finite trees. Mathematical Proceedings of the Cambridge Philosophical Society 59(4), 833–835 (1963)

    Article  MathSciNet  Google Scholar 

  40. Ozbagci, B., Popescu-Pampu, P.: Generalized plumbings and murasugi sums. Arnold Mathematical Journal 2(1), 69–119 (2015)

    Article  MathSciNet  Google Scholar 

  41. Piccirillo, L.: The conway knot is not slice. Ann. Math. 191(2), 581–591 (2020)

    Article  MathSciNet  Google Scholar 

  42. Rolfsen, D.: Knots and links. AMS Chelsea Pub, Providence, R.I (2003). OCLC: ocm52901393

  43. Rudolph, L.: Quasipositivity as an obstruction to sliceness. Bull. Am. Math. Soc. 29(1), 51–59 (1993)

    Article  MathSciNet  Google Scholar 

  44. Rudolph, L.: Positive links are strongly quasipositive. Geometry & Topology Monographs, Volume 2: Proceedings of the Kirbyfest, pages 555–562 (1998)

  45. Sakuma, M.: Minimal genus seifert surfaces for special arborescent links. Osaka Journal of Mathematics 31, 861–905 (1994)

    MathSciNet  Google Scholar 

  46. Stallings, J.R.: Constructions of fibred knots and links. Proc. Symp. Pure Math., AMS 27, 315–319 (1975)

    Google Scholar 

  47. Teichner, P.: Slice knots: Knot theory in the 4th dimension, 2011. Lecture notes by Julia Collins and Mark Powell. Electronic version available from https://www.maths.ed.ac.uk/~v1ranick/papers/sliceknots2.pdf

  48. Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. 87, 56–88 (1968)

    Article  MathSciNet  Google Scholar 

  49. Weinberger, S.: Homology manifolds. Handbook of geometric topology, 1085–1102 (2002)

  50. Whitten, W.: Isotopy types of knot spanning surface. Topology 12, 373–380 (1973)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank Sebastian Baader for helpful discussions, and the anonymous reviewers for their questions and suggestions which allowed us to significantly improve the paper.

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The authors did not receive support from any organization for the submitted work.

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Authors and Affiliations

  1. Aix-Marseille Université, CNRS, I2M, Marseille, France

    Pierre Dehornoy

  2. Univ Gustave Eiffel, CNRS, LIGM, F-77454, Marne-la-Vallée, France

    Corentin Lunel & Arnaud de Mesmay

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  1. Pierre Dehornoy
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  2. Corentin Lunel
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Dehornoy, P., Lunel, C. & de Mesmay, A. Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect. Discrete Comput Geom (2026). https://doi.org/10.1007/s00454-025-00788-5

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