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Admissible transverse surgery does not preserve tightness

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Abstract

We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots—namely, the range of slopes on which admissible transverse surgery preserves tightness—and to provide some new examples of knot types which are not uniformly thick. Our examples also illuminate several interesting new phenomena, including the existence of hyperbolic, universally tight contact 3-manifolds whose Heegaard Floer contact invariants vanish (and which are not weakly fillable); and the existence of open books with arbitrarily high fractional Dehn twist coefficients whose compatible contact structures are not deformations of co-orientable taut foliations.

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References

  1. Agol, I.: Bounds on exceptional Dehn filling. Geom. Topol. 4, 431–449 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baldwin, J.A.: Tight contact structures and genus one fibered knots. Algebr. Geom. Topol. 7, 701–735 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baldwin, J.A.: Capping off open books and the Ozsváth–Szabó contact invariant. Math. Res. Lett. 19(1), 31–40 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berge, J.: The knots in \(D^2\times S^1\) which have nontrivial Dehn surgeries that yield \(D^2\times S^1\). Topol. Appl. 38(1), 1–19 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Colin, V., Honda, K.: Reeb vector fields and open book decompositions (2008). math.GT/0809.5088

  6. Colin, V., Honda, K., Ghiggini, P.: The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions (2013, in preparation)

  7. Colin, V., Honda, K., Ghiggini, P.: Embedded contact homology and open book decompositions (2010). math.SG/1008.2734

  8. Colin, V.: Chirurgies de Dehn admissibles dan les variétés de contact tendues. Ann. Inst. Fourier 51(5), 1419–1435 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Eliashberg, Y.: Invariants in contact topology. In: Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Documenta Mathematica, Extra vol. II, pp. 327–338 (1998)

  11. Eliashberg, Y.: A few remarks about symplectic fillings. Geom. Topol. 8, 277–293 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eliashberg, Y., Thurston, W.: Confoliations. University Lecture Series, vol. 13. American Mathematical Society, Providence (1998)

  13. Etnyre, J.B.: Planar open book decompositions and contact structures. Int. Math. Res. Not. 2004(79), 4255–4267 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Etnyre, J.B.: On symplectic fillings. Algebr. Geom. Topol. 4, 73–80 (2004) (electronic)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Etnyre, J.B., Honda, K.: Tight contact structures with no symplectic fillings. Invent. Math. 148, 609–626 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Etnyre, J.B., Honda, K.: Cabling and transverse simplicity. Ann. Math. (2), 162(3), 1305–1333 (2005)

    Google Scholar 

  18. Gabai, D.: Surgery on knots in solid tori. Topology 28(1), 1–6 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gay, D.: Symplectic 4-dimensional 2-handles and contact surgery along transverse knots. PhD thesis, Berkeley (1999)

  20. Gay, D.: Four-dimensional symplectic cobordisms containing three-handles. Geom. Topol. 10, 1749–1759 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Geiges, H.: Constructions of contact manifolds. Math. Proc. Camb. Philos. Soc. 121(3), 455–464 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ghiggini, P.: Ozsváth–Szabó invariants and fillability of contact structures (2005) math.GT/0403367

  23. Ghiggini, P., Honda, K.: Giroux torsion and twisted coefficients (2008). math.GT/0804.1568

  24. Ghiggini, P., Honda, K., Van Horn-Morris, J.: The vanishing of the contact invariant in the presence of torsion (2007). math.GT/0706.1602

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Honda, K.: Gluing tight contact structures. Duke Math. J. 115(3), 435–478 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Honda, K., Kazez, W., Matić, G.: Convex decomposition theory. Int. Math. Res. Not. 2002(2), 55–88 (2002)

    Article  MATH  Google Scholar 

  30. Honda, K., Kazez, W., Matić, G.: Right-veering diffeomorphisms of a compact surface with boundary. Invent. Math. 169(2), 427–449 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Honda, K., Kazez, W., Matić, G.: Right-veering diffeomorphisms of a compact surface with boundary II. Geom. Topol. 12, 2057–2094 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kanda, Y.: The classification of tight contact structures on the \(3\)-torus. Commun. Anal. Geom. 5(3), 413–438 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Kutluhan, C., Lee, Y.-J., Taubes, C.H.: HF = HM I: Heegaard Floer homology and Seiberg-Witten Floer homology (2010). math.SG/1007.1979

  34. Kutluhan, C., Lee, Y.J., Taubes, C.H.: HF = HM II: Reeb orbits and holomorphic curves for the ech/Heegaard-Floer correspondence (2010). math.SG/1008.1595

  35. Kutluhan, C., Lee, Y.J., Taubes, C.H.: HF = HM III: holomorphic curves and the differential for the ech/Heegaard-Floer correspondence (2010). math.SG/1010.3456

  36. Lekili, Y., Ozbagci, B.: Milnor fillable contact structures are universally tight. Math. Res. Lett. 17(5–6), 1055–1064 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. LaFountain, D.J.: Studying uniform thickness I: Legendrian simple iterated torus knots. Algebr. Geom. Topol. 10(2), 891–916 (2010) (electronic)

    Google Scholar 

  38. LaFountain, D.J.: Studying uniform thickness II: transversely non-simple iterated torus knots (2009). e-print at arxiv: 0909.1452

  39. Latschev, J., Wendl, C.: Algebraic torsion in contact manifolds. Geom. Funct. Anal. 21(5), 1144–1195 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lerman, E.: Contact cuts. Isr. J. Math. 124, 77–92 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  42. Massot, P.: Infinitely many universally tight torsion free contact manifolds with vanishing Ozsváth–Szabó contact invariants (2009). math.GT/0912.5107

  43. Neiderkrüger, K., Wendl, C.: Weak symplectic fillings and holomorphic curves (2010). math.SG/1003.3923

  44. Ohta, H., Ono, K.: Simple singularities and topology of symplectically filling 4-manifold. Comment. Math. Helv. 74, 575–590 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ozsváth, P., Szabó, Z.: Heegaard Floer homologies and contact structures. Duke Math. J. 129(1), 39–61 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  46. Penner, R.C.: A construction of pseudo-Anosov homeomorphisms. Trans. Am. Math. Soc. 310(1), 179–197 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  47. Plamenevskaya, O., Van Horn-Morris, J.: Planar open books, monodromy factorizations and symplectic fillings. Geom. Topol. 14, 2077–2101 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Roberts, R.: Taut foliations in punctured surface bundles, II. Proc. Lond. Math. Soc. 3(83), 443–471 (2001)

    Google Scholar 

  49. Thurston, W.: The Geometry and Topology of Three-Manifolds. Princeton, NJ (1979)

  50. Thurston, W.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. 19(2), 417–431 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wendl, C.: A hierarchy of local symplectic filling obstructions for, contact 3-manifolds (2010). math.SG/1009.2746

  52. Wendl, C.: Non-exact symplectic cobordisms between, contact 3-manifolds (2010). math.SG/1008.2456

  53. Yau, M.L.: Vanishing of the contact homology of overtwisted, contact 3-manifolds (2004). math.SG/0411014

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Acknowledgments

The authors thank Vincent Colin, Tobias Ekholm, Yasha Eliashberg, Ko Honda, Janko Latschev and Chris Wendl for helpful correspondence. JAB was partially supported by NSF Grant DMS-1104688 and JBE was partially supported by NSF Grant DMS-0804820 and thanks the University of Texas, Austin for its hospitality while working on parts of this paper.

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Baldwin, J.A., Etnyre, J.B. Admissible transverse surgery does not preserve tightness. Math. Ann. 357, 441–468 (2013). https://doi.org/10.1007/s00208-013-0911-8

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