Skip to main content
SpringerLink
Log in

On cobordisms between knots, braid index, and the Upsilon-invariant

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We use Ozsváth, Stipsicz, and Szabó’s Upsilon-invariant to provide bounds on cobordisms between knots that ‘contain full-twists’. In particular, we recover and generalize a classical consequence of the Morton–Franks–Williams inequality for knots: positive braids that contain a positive full-twist realize the braid index of their closure. We also establish that quasi-positive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure. Finally, we provide inductive formulas for the Upsilon-invariant of torus knots and compare it to the Levine–Tristram signature profile.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. Strictly speaking this is not a generalization since our setting is restricted to knots, while the original result holds for all links.

  2. While the application of [26, Theorem 1.15] yields the result stated here, the preprint available as of this writing contains a small typo in Proposition 6.3 (the index i is shifted by 1), which accounts for the discrepancy between the result quoted here and that written in [26].

References

  1. Alexander, J.W.: A lemma on systems of knotted curves. Proc. Nat. Acad. Sci. USA 9, 93–95 (1923)

    Article  Google Scholar 

  2. Artin, E.: Theorie der Zöpfe. Abh. Math. Sem. Univ. Hambg. 4(1), 47–72 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borodzik, M., Hedden, M.: The Upsilon Function of L-Space Knots is a Legendre Transform (2015). arXiv e-prints: arXiv:1505.06672 [math.GT]

  4. Borodzik, M., Livingston, C.: Semigroups, \(d\)-invariants and deformations of cuspidal singular points of plane curves. J. Lond. Math. Soc. (2) 93(2), 439–463 (2016). arXiv:1305.2868 [math.AG]

  5. Bodnár, J., Némethi, A.: Lattice cohomology and rational cuspidal curves. Math. Res. Lett 23(2), 339–375 (2016). arXiv:1405.0437 [math.AG]

  6. Boileau, M., Orevkov., S.: Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math. 332(9), 825–830 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brandenbursky, M.: On quasi-morphisms from knot and braid invariants. J. Knot Theory Ramif. 20(10), 1397–1417 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dore, D.: Recursive Properties of the Upsilon Invariant for Torus Knots. Princeton Undergraduate Thesis (2015)

  9. Dynnikov, I.A., Prasolov, M.V.: Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions. Trans. Mosc. Math. Soc. 74(1), 97–144 (2013)

  10. Feller, P.: Optimal cobordisms between torus knots. Commun. Anal. Geom. 24(5), 993–1025 (2016). arXiv:1501.00483 [math GT]

  11. Feller, P: A Sharp Signature Bound for Positive Four-Braids (2015). arXiv e-prints: arXiv:1508.00418 [math.GT]

  12. Franks, J., Williams, R.F.: Braids and the Jones polynomial. Trans. Am. Math. Soc. 303(1), 97–108 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gambaudo, J.-M., Ghys, É.: Braids and signatures. Bull. Soc. Math. Fr. 133(4), 541–579 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gordon, C.McA, Litherland, R.A., Murasugi, K.: Signatures of covering links. Can. J. Math. 33(2), 381–394 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hayden, K.: Minimal Braid Representatives of Quasipositive links (2016). arXiv e-prints: ArXiv:1605.01711 [math.GT]

  16. Hedden, M., Watson, L.: On the Geography and Botany of Knot Floer Homology (2014). arXiv e-prints: arXiv:1404.6913 [math.GT]

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Krcatovich, D.: A Restriction on the Alexander Polynomials of L-Space Knots (2014). arXiv e-prints: arXiv:1408.3886

  19. Levine, J.: Knot cobordism groups in codimension two. Comment. Math. Helv. 44, 229–244 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  20. Livingston, Charles: Computations of the Ozsváth-Szabó knot concordance invariant. Geom. Topol. 8, 735–742 (2004) (electronic). arXiv:0311036v3 [math.GT]

  21. LaFountain, D.J., Menasco, W.W.: Embedded annuli and Jones’ conjecture. Algebr. Geom. Topol. 14(6), 3589–3601 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Morton, H.R.: Seifert circles and knot polynomials. Math. Proc. Camb. Philos. Soc. 99(1), 107–109 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Murasugi, K.: On closed \(3\)-braids. Memoirs of the American Mathematical Society, No. 151. American Mathematical Society, Providence (1974)

  24. Ozsváth, P., Szabó, Z.: Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math. 173(2), 179–261 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ozsváth, P., Szabó, Z.: Knot Floer homology and the four-ball genus. Geom. Topol. 7, 615–639 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ozsváth, P., Stipsicz, A.I., Szabó, Z.: Concordance Homomorphisms from Knot Floer Homology (2014). arXiv e-prints: arXiv:1407.1795 [math.GT]

  27. Ozsváth, P., Stipsicz, A.I., Szabó, Z.: Unoriented Knot Floer Homology and the Unoriented Four-Ball Genus (2015). arXiv e-prints: arXiv:1508.03243 [math.GT]

  28. Rudolph, L.: Algebraic functions and closed braids. Topology 22(2), 191–202 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Stoimenow, A.: On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks. Trans. Am. Math. Soc 354(10), 3927–3954 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Stoimenow, A.: Bennequin’s inequality and the positivity of the signature. Trans. Am. Math. Soc. 360(10), 5173–5199 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Tristram, A.G.: Some cobordism invariants for links. Proc. Camb. Philos. Soc. 66, 251–264 (1969)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Wall, C.T.C.: Singular Points of Plane Curves. London Mathematical Society Students Texts, vol. 63. Cambridge University Press, Cambridge (2004)

  35. Wang, S.: On the first singularity for the upsilon invariant of algebraic knots. Bull. Lond. Math. Soc 48(2), 349–354 (2016). arXiv:1505.06835 [math.GT]

Download references

Acknowledgements

We thank Sebastian Baader, Matt Hedden, Lukas Lewark, and Aru Ray for helpful discussions. Thanks also to Peter Ozsváth for pointing us to Dan Dore’s [8]. We owe special thanks to Maciej Borodzik, who referred us to [5, Proposition 5.2.4], which we need to compute \(\Upsilon \) of torus knots in full generality. Finally, many thanks to the anonymous referee for their detailed and on point suggestions.

Author information

Authors and Affiliations

  1. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111, Bonn, Germany

    Peter Feller

  2. Department of Mathematics, Rice University, Houston, TX, 77047, USA

    David Krcatovich

Authors
  1. Peter Feller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Krcatovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Krcatovich.

Additional information

Peter Feller gratefully acknowledges support by the Swiss National Science Foundation Grant 155477. David Krcatovich was partially supported by NSF Grant DMS-1309081.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feller, P., Krcatovich, D. On cobordisms between knots, braid index, and the Upsilon-invariant. Math. Ann. 369, 301–329 (2017). https://doi.org/10.1007/s00208-017-1519-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-017-1519-1

Mathematics Subject Classification

Search

Navigation