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Stein traces and characterizing slopes

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Abstract

We show that there exists an infinite family of pairwise non-isotopic Legendrian knots in the standard contact 3-sphere whose Stein traces are equivalent. This is the first example of such phenomenon. Different constructions are developed in the article, including a contact annulus twist, explicit Weinstein handlebody equivalences, and a discussion on dualizable patterns in the contact setting. These constructions can be used to systematically construct distinct Legendrian knots in the standard contact 3-sphere with contactomorphic \((-1)\)-surgeries and, in many cases, equivalent Stein traces. In addition, we also discuss characterizing slopes and provide results in the opposite direction, i.e. describe cases in which the Stein trace, or the contactomorphism type of an r-surgery, uniquely determines the Legendrian isotopy type.

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Notes

  1. The n-trace of a knot K in \( S^3\) is the smooth 4-manifold obtained as the result of attaching a 2-handle to the 3-sphere boundary of the 4-ball along the knot K with framing n.

  2. We denote the natural numbers including zero by \({\mathbb {N}}_0\) and the natural numbers without zero by \({\mathbb {N}}\).

  3. For reference, we note that L(r) consists of a unique contact manifold if and only if \(r=1/n\) for some integer n. So our notation for \(L(\pm 1)\) agrees with the notation above. If \(n>1\) is an integer, then L(n) consists of two contact manifolds, and if n is a negative integer, then L(n) consists of |n| contact manifolds, which are not always different.

  4. We observe that the question which contact surgery coefficients are characterizing differs from the topological question since we have always infinitely many Legendrian realizations of a single topological knot type and the framings are measured with respect to the contact framing.

  5. Here our notion of connected sum is an internal one happening in the surgered manifold. We will explain this in detail in the proof and also argue why J represents a standard Legendrian unknot in \(L(\pm 1/n)\) and thus the connected sum is well-defined.

References

  1. Abe, T., Jong, I.D., Luecke, J., Osoinach, J.: Infinitely many knots admitting the same integer surgery and a four-dimensional extension. Int. Math. Res. Not. IMRN 22, 11667–11693 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Abe, T., Jong, I.D., Omae, Y., Takeuchi, M.: Annulus twist and diffeomorphic 4-manifolds. Math. Proc. Camb. Philos. Soc. 155(2), 219–235 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Akbulut, S.: On \(2\)-dimensional homology classes of \(4\)-manifolds. Math. Proc. Camb. Philos. Soc. 82(1), 99–106 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnol’d, V.I., Givental’, A.B.: Symplectic geometry. In: Dynamical Systems, IV, Encyclopaedia Math. Sci., vol. 4, pp. 1–138. Springer, Berlin (2001)

  5. Avdek, R.: Contact surgery and supporting open books. Algebr. Geom. Topol. 13(3), 1613–1660 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baker, K.L., Gordon, C., Luecke, J.: Bridge number and integral Dehn surgery. Algebr. Geom. Topol. 16(1), 1–40 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baker, K.L., Motegi, K.: Noncharacterizing slopes for hyperbolic knots. Algebr. Geom. Topol. 18(3), 1461–1480 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bourgeois, F., Ekholm, T., Eliashberg, Y.: Effect of Legendrian surgery. Geom. Topol. 16(1), 301–389 (2012). With an appendix by Sheel Ganatra and Maksim Maydanskiy

  9. Brakes, W.R.: Manifolds with multiple knot-surgery descriptions. Math. Proc. Camb. Philos. Soc. 87(3), 443–448 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Casals, R., Etnyre, J.B., Kegel, M.: Stein Traces and Characterizing Slopes. arXiv version arXiv:2111.00265

  11. Chakraborty, A., Etnyre, J.B., Min, H.: Cabling legendrian and transverse knots (2020)

  12. Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and back, American Mathematical Society Colloquium Publications, vol. 59. American Mathematical Society, Providence. Symplectic geometry of affine complex manifolds (2012)

  13. Conway, J., Etnyre, J.B., Tosun, B.: Symplectic fillings, contact surgeries, and Lagrangian disks. Int. Math. Res. Not. IMRN 8, 6020–6050 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Culler, M., Dunfield, N., Goerner, M., Weeks, J.: Snappy, a computer program for studying the geometry and topology of 3-manifolds

  15. Culler, M., Gordon, C.McA., Luecke, J., Shalen, P.B.: Dehn surgery on knots. Ann. Math. (2) 125(2), 237–300 (1987)

  16. Ding, F., Geiges, H.: A Legendrian surgery presentation of contact 3-manifolds. Math. Proc. Camb. Philos. Soc. 136(3), 583–598 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ding, F., Geiges, H.: Handle moves in contact surgery diagrams. J. Topol. 2(1), 105–122 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ding, F., Geiges, H., Stipsicz, A.I.: Surgery diagrams for contact 3-manifolds. Turk. J. Math. 28(1), 41–74 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Durst, S., Kegel, M.: Computing rotation and self-linking numbers in contact surgery diagrams. Acta Math. Hungar. 150(2), 524–540 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Eliashberg, Y., Fraser, M.: Classification of topologically trivial Legendrian knots. In: Geometry, Topology, and Dynamics (Montreal, PQ, 1995), CRM Proc. Lecture Notes, vol. 15, pp. 17–51. Amer. Math. Soc., Providence (1998)

  21. Eliashberg, Y.: Classification of overtwisted contact structures on \(3\)-manifolds. Invent. Math. 98(3), 623–637 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. Etnyre, J.B., Honda, K.: Knots and contact geometry. I. Torus knots and the figure eight knot. J. Symplectic Geom. 1(1), 63–120 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Etnyre, J.B., Kegel, M., Onaran, S.: Contact surgery numbers, Preprint (2021)

  24. Etnyre, J.B., LaFountain, D.J., Tosun, B.: Legendrian and transverse cables of positive torus knots. Geom. Topol. 16(3), 1639–1689 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Etnyre, J.B., Ng, L.L., Vértesi, V.: Legendrian and transverse twist knots. J. Eur. Math. Soc. (JEMS) 15(3), 969–995 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Etnyre, J.B.: On contact surgery. Proc. Am. Math. Soc. 136(9), 3355–3362 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gabai, D.: Foliations and the topology of \(3\)-manifolds. II. J. Differ. Geom. 26(3), 461–478 (1987)

    MathSciNet  MATH  Google Scholar 

  28. Geiges, H.: An introduction to contact topology. Cambridge Studies in Advanced Mathematics, vol. 109. Cambridge University Press, Cambridge (2008)

  29. Ghiggini, P.: Knot Floer homology detects genus-one fibred knots. Am. J. Math. 130(5), 1151–1169 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Giroux, E.: Structures de contact en dimension trois et bifurcations des feuilletages de surfaces. Invent. Math. 141(3), 615–689 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gompf, R.E.: Handlebody construction of Stein surfaces. Ann. Math. (2) 148(2), 619–693 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

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

  34. Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  35. Honda, K.: On the classification of tight contact structures. I. Geom. Topol. 4, 309–368 (2000) (electronic)

  36. Kegel, M.: The Legendrian knot complement problem. J. Knot Theory Ramif. 27(14), 1850067 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kirby, R.: Problems in low dimensional manifold theory. In: Algebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pp. 273–312. Amer. Math. Soc., Providence (1978)

  38. Kronheimer, P., Mrowka, T., Ozsváth, P., Szabó, Z.: Monopoles and lens space surgeries. Ann. Math. (2) 165(2), 457–546 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lackenby, M.: Every knot has characterising slopes. Math. Ann. 374(1–2), 429–446 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lazarev, O.: Geometric and algebraic presentations for Weinstein domains (2019)

  41. Manolescu, C., Piccirillo, L.: From zero surgeries to candidates for exotic definite four-manifolds (2021)

  42. McCoy, D.: On the characterising slopes of hyperbolic knots. Math. Res. Lett. 26(5), 1517–1526 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. McCoy, D.: Non-integer characterizing slopes for torus knots. Commun. Anal. Geom. 28(7), 1647–1682 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. McDuff, D.: The structure of rational and ruled symplectic \(4\)-manifolds. J. Am. Math. Soc. 3(3), 679–712 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  45. Miller, A.N., Piccirillo, L.: Knot traces and concordance. J. Topol. 11(1), 201–220 (2018)

  46. Moser, L.: Elementary surgery along a torus knot. Pac. J. Math. 38, 737–745 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  47. Neumann, W.D., Zagier, D.: Volumes of hyperbolic three-manifolds. Topology 24(3), 307–332 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ni, Y., Zhang, X.: Characterizing slopes for torus knots. Algebr. Geom. Topol. 14(3), 1249–1274 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. Osoinach, J.K., Jr.: Manifolds obtained by surgery on an infinite number of knots in \(S^3\). Topology 45(4), 725–733 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  50. Ozsváth, P., Szabó, Z.: The Dehn surgery characterization of the trefoil and the figure eight knot. J. Symplectic Geom. 17(1), 251–265 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  51. Piccirillo, L.: Shake genus and slice genus. Geom. Topol. 23(5), 2665–2684 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  52. Tagami, K.: Notes on constructions of knots with the same trace (2020)

  53. Tagami, K.: On annulus presentations, dualizable patterns and rgb-diagrams (2020)

  54. Thurston, W.P.: The geometry and topology of 3-manifolds. Princeton Lecture Notes (1980)

  55. Weinstein, A.: Contact surgery and symplectic handlebodies. Hokkaido Math. J. 20(2), 241–251 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Bob Gompf, Oleg Lazarev, and Lisa Piccirillo for helpful comments on an earlier draft of this paper. We also thank the anonymous referee for many helpful comments and suggestions that have improved the paper. Roger Casals is supported by the NSF CAREER Award DMS-1942363 and the Alfred P. Sloan Foundation. He is also thankful to John Etnyre and the Georgia Institute of Technology for their hospitality during his May 2019 visit, where we first started to discuss this project. John Etnyre thanks Lisa Piccirillo for very helpful conversations about the annulus twist and a beautiful set of lectures at the 2021 Tech Topology Summer School that informed our understanding of some of the constructions in Sect. 3.5. John Etnyre was partially supported by NSF grant DMS-1906414 and DMS-2203312. Marc Kegel would like to thank the Mathematisches Forschungsinstitut Oberwolfach where large parts of this project were carried out as a Oberwolfach Research Fellow in August 2020.

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Authors and Affiliations

  1. Department of Mathematics, University of California Davis, Shields Avenue, Davis, CA, 95616, USA

    Roger Casals

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

    John Etnyre

  3. Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Marc Kegel

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  1. Roger Casals
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Correspondence to John Etnyre.

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Casals, R., Etnyre, J. & Kegel, M. Stein traces and characterizing slopes. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02662-2

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