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On classical upper bounds for slice genera

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Abstract

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.

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References

  1. Baader, S.: On the stable 4-genus of knots with indefinite Seifert form. Commun. Anal. Geom. 24(2), 301–305 (2016). arXiv:1408.6091

    Article  MathSciNet  Google Scholar 

  2. 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). arXiv:1509.07634

    Article  MathSciNet  Google Scholar 

  3. Baader, S., Lewark, L.: The stable 4-genus of alternating knots. Asian J. Math. (6), 1183–1190 (2017). arXiv:1505.03345

  4. Borodzik, M., Friedl, S.: Knotorious world wide web page. http://www.mimuw.edu.pl/~mcboro/knotorious.php, retrieved November 8 (2016)

  5. Borodzik, M., Friedl, S.: On the algebraic unknotting number. Trans. Lond. Math. Soc. 1(1), 57–84 (2014). arXiv:1308.6105

    Article  MathSciNet  Google Scholar 

  6. Borodzik, M., Friedl, S.: The unknotting number and classical invariants, I. Algebr. Geom. Topol. 15(1), 85–135 (2015). arXiv:1203.3225

    Article  MathSciNet  Google Scholar 

  7. Casson, A. J., McA, C.: Gordon: on slice knots in dimension three, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, Am. Math. Soc., Providence, RI (1978) pp. 39–53

  8. Casson, A. J., Gordon, C. McA.: Cobordism of classical knots, À la recherche de la topologie perdue, Progr. Math., vol. 62, Birkhäuser Boston, Boston, MA, With an appendix by P. M. Gilmer, pp. 181–199 (1986)

  9. Cha, J.C.: Topological minimal genus and \(L^2\)-signatures. Algebr. Geom. Topol. 8(2), 885–909 (2008)

    Article  MathSciNet  Google Scholar 

  10. Cha, J. C., Livingston, C.: KnotInfo: Table of Knot Invariants. www.indiana.edu/~knotinfo, retrieved on November 8 (2016)

  11. Cochran, T. D., Orr, K. E., Teichner, P.: Knot concordance, Whitney towers and \(L^2\)-signatures, Ann. of Math. (2) 157(2), pp. 433–519 (2003). arXiv:math/9908117

  12. Feller, P.: The degree of the Alexander polynomial is an upper bound for the topological slice genus. Geom. Topol. 20(3), 1763–1771 (2016). arXiv:1504.01064

    Article  MathSciNet  Google Scholar 

  13. Feller, P., McCoy, D.: On 2-bridge knots with differing smooth and topological slice genera. Proc. Am. Math. Soc. 144(12), 5435–5442 (2016). arXiv:1508.01431

    Article  MathSciNet  Google Scholar 

  14. Fogel, M.: The algebraic unknotting number, Ph.D. thesis, University of California, Berkeley (1993)

  15. Freedman, M.H.: The topology of four-dimensional manifolds. J. Differ. Geom. 17(3), 357–453 (1982)

    Article  MathSciNet  Google Scholar 

  16. Freedman, M.H., Quinn, F.: Topology of 4-manifolds, Princeton Mathematical Series, vol. 39. Princeton University Press, Princeton, NJ (1990)

    Google Scholar 

  17. Garoufalidis, S., Teichner, P.: On knots with trivial Alexander polynomial. J. Differ. Geom. 67(1), 167–193 (2004)

    Article  MathSciNet  Google Scholar 

  18. Gilmer, P.M.: On the slice genus of knots. Invent. Math. 66(2), 191–197 (1982)

    Article  MathSciNet  Google Scholar 

  19. Gompf, R.E., Stipsicz, A.I.: \(4\)-manifolds and Kirby Calculus, Graduate Studies in Mathematics, vol. 20. American Mathematical Society, Providence, RI (1999)

    MATH  Google Scholar 

  20. Kim, T.: An infinite family of non-concordant knots having the same Seifert form. Comment. Math. Helv. 80(1), 147–155 (2005). arXiv:math/0402425

  21. Ko, K.H.: A Seifert-matrix interpretation of Cappell and Shaneson’s approach to link cobordisms. Math. Proc. Cambridge Philos. Soc. 106(3), 531–545 (1989)

    Article  MathSciNet  Google Scholar 

  22. Kronheimer, P.B., Mrowka, T.S.: Gauge theory for embedded surfaces, I. Topology 32(4), 773–826 (1993)

    Article  MathSciNet  Google Scholar 

  23. Lewark, L., McCoy, D.: On calculating the slice genera of 11- and 12-crossing knots, accepted for publication by Exp. Math. (2016) arXiv:1508.01098v2

  24. Liechti, L.: Positive braid knots of maximal topological 4-genus. Math. Proc. Camb. Philos. Soc. 161(3), 559–568 (2016). arXiv:1511.03883

    Article  MathSciNet  Google Scholar 

  25. Litherland,R. A.: Signatures of iterated torus knots, topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, : Lecture Notes in Math., vol. 722. Springer, Berlin 1979, 71–84 (1977)

    Google Scholar 

  26. Livingston,C.: Examples in Concordance, a shortened version of this article has been published as [27] (2001). arXiv:math/0101035v2

  27. Livingston, C.: Seifert forms and concordance. Geom. Topol. 6, 403–408 (2002). arXiv:math/0101035v3

    Article  MathSciNet  Google Scholar 

  28. Livingston, C.: The stable 4-genus of knots. Algebr. Geom. Topol. 10(4), 2191–2202 (2010). arXiv:0904.3054

    Article  MathSciNet  Google Scholar 

  29. Livingston, C., Naik, S.: Ozsváth-Szabó and Rasmussen invariants of doubled knots. Algebr. Geom. Topol. 6, 651–657 (2006). arXiv:math/0505361

    Article  MathSciNet  Google Scholar 

  30. Milnor, J., Husemöller,D.: Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, Springer (1973)

  31. Murakami,H.: Algebraic unknotting operation, Proceedings of the Second Soviet-Japan Joint Symposium of Topology (Khabarovsk, 1989), vol. 8, Questions Answers Gen. Topology, vol. 1, pp. 283–292 (1990)

  32. Rolfsen,D.: Knots and Links, Mathematics Lecture Series, vol. 7, Publish or Perish Inc., Houston, TX, Corrected reprint of the 1976 original (1990)

  33. Rudolph, L.: Constructions of quasipositive knots and links, I, Knots, Braids and Singularities (Plans-sur-Bex, : Monogr. Enseign. Math., vol. 31, Enseignement Math. Geneva 1983, 233–245 (1982)

  34. Rudolph, L.: Some topologically locally-flat surfaces in the complex projective plane. Comment. Math. Helv. 59(4), 592–599 (1984)

    Article  MathSciNet  Google Scholar 

  35. Rudolph,L.: Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. (N.S.) 29(1), 51–59 (1993). arXiv:math/9307233

    Article  MathSciNet  Google Scholar 

  36. Saeki, O.: On algebraic unknotting numbers of knots. Tokyo J. Math. 22(2), 425–443 (1999)

    Article  MathSciNet  Google Scholar 

  37. Taylor,L. R.: On the genera of knots, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, : Lecture Notes in Math., vol. 722. Springer, Berlin 1979, 144–154 (1977)

    Google Scholar 

  38. Trotter, H.F.: Homology of group systems with applications to knot theory. Ann. Math. 76(2), 464–498 (1962)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We thank Danny Ruberman for pointing us to [37]. We thank Sebastian Baader and Livio Liechti for valuable inputs; in particular, concerning Proposition 25. We thank Mark Powell for comments on a first version of this paper, and the referee for helpful suggestions. Both authors gratefully acknowledge support by the SNSF and thank the MPIM Bonn for its hospitality.

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  1. Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Peter Feller

  2. Mathematical Institute, University of Bern, Alpeneggstrasse 22, 3012, Bern, Switzerland

    Lukas Lewark

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  1. Peter Feller
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  2. Lukas Lewark
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Feller, P., Lewark, L. On classical upper bounds for slice genera. Sel. Math. New Ser. 24, 4885–4916 (2018). https://doi.org/10.1007/s00029-018-0435-x

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