Skip to main content
SpringerLink
Log in

Essential open book foliations and fractional Dehn twist coefficient

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We introduce essential open book foliations by refining open book foliations, and develop technical estimates of the fractional Dehn twist coefficient (FDTC) of monodromies and the FDTC for closed braids, which we introduce as well. As applications, we quantitatively study the ‘gap’ between overtwisted contact structures and non-right-veering monodromies. We give sufficient conditions for a 3-manifold to be irreducible and atoroidal. We also show that the geometries of a 3-manifold and the complement of a closed braid are determined by the Nielsen–Thurston types of the monodromies of their open book decompositions.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33

Similar content being viewed by others

References

  1. Bennequin, D.: Entrelacements et équations de Pfaff. Astérisque 107–108, 87–161 (1983)

    MathSciNet  Google Scholar 

  2. Birman, J., Finkelstein, E.: Studying surfaces via closed braids. J. Knot Theory Ramif. 7(3), 267–334 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Birman, J., Menasco, W.: Studying links via closed braids. IV. Composite links and split links. Invent. Math. 102(1), 115–139 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Birman, J., Menasco, W.: Studying links via closed braids. II. On a theorem of Bennequin. Topol. Appl. 40(1), 71–82 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Birman, J., Menasco, W.: Studying links via closed braids. V. The unlink. Trans. Am. Math. Soc. 329(2), 585–606 (1992)

    MathSciNet  MATH  Google Scholar 

  6. Birman, J., Menasco, W.: Studying links via closed braids. I. A finiteness theorem. Pac. J. Math. 154(1), 17–36 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Birman, J., Menasco, W.: Studying links via closed braids. VI. A nonfiniteness theorem. Pac. J. Math. 156(2), 265–285 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Birman, J., Menasco, W.: Studying links via closed braids. III. Classifying links which are closed 3-braids. Pac. J. Math. 161(1), 25–113 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Birman, J., Menasco, W.: Special positions for essential tori in link complements. Topology 33(3), 525–556 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Birman, J., Menasco, W.: Stabilization in the braid groups. I. MTWS. Geom. Topol. 10, 413–540 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Birman, J., Menasco, W.: Stabilization in the braid groups. II. Transversal simplicity of knots. Geom. Topol. 10, 1425–1452 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Colin, V., Honda, K.: Stabilizing the monodromy of an open book decomposition. Geom. Dedicata. 132, 95–103 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Colin, V., Honda, K.: Reeb vector fields and open book decompositions. J. Eur. Math. Soc. (JEMS) 15(2), 443–507 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eliashberg, Y.: Contact 3-manifolds twenty years since J. Martinet’s work. Ann. Inst. Fourier (Grenoble) 42, 165–192 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word Processing in Groups. Jones and Bartlett Publishers, Boston (1992)

    MATH  Google Scholar 

  16. Etnyre, J., Ozbagci, B.: Invariants of contact structures from open books. Trans. Am. Math. Soc. 360(6), 3133–3151 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton University Press, Princeton (2011)

    Book  MATH  Google Scholar 

  18. Gabai, D., Kazez, W.: Detecting fibred links in \(S^{3}\). Comment. Math. Helv. 61, 519–555 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gabai, D., Kazez, W.: Group negative curvature for 3-manifolds with genuine laminations. Geom. Topol. 2, 65–77 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gabai, D., Oertel, U.: Essential laminations in 3-manifolds. Ann. Math. 130, 41–73 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Geiges, H.: An Introduction to Contact Topology. Cambridge Studies in Advanced Mathematics, vol. 109. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  22. Giroux, E.: Géométrie de contact: de la dimension trois vres les dimensions supérieures. In: Proceedings of the International Congress of Mathematics, vol. II (Beijing, 2002), pp. 405-414

  23. Gordon, C.: Dehn surgery and 3-manifolds. Low dimensional topology. IAS/Park City Math. Ser., 15, Am. Math. Soc., Providence, RI, pp. 21–71(2009)

  24. Handel, M., Thurston, W.P.: New proofs of some results of Nielsen. Adv. Math. 56(2), 173–191 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hatcher, A.: The Classification of \(3\)-Manifolds—A Brief Overview. Available at http://www.math.cornell.edu/~hatcher/Papers/3Msurvey

  26. Honda, K., Kazez, W., Matić, G.: Right-veering diffeomorphisms of compact surfaces with boundary. Invent. math. 169(2), 427–449 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ito, T.: Braid ordering and the geometry of closed braid. Geom. Topol. 15, 473–498 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ito, T.: Braid ordering and knot genus. J. Knot Theory Ramif. 20, 1311–1323 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ito, T., Kawamuro, K.: Open book foliations. Geom. Topol. 18, 1581–1634 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ito, T., Kawamuro, K.: Visualizing overtwisted discs in open books. Publ. RIMS Kyoto Univ. 50, 169–180 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ito, T., Kawamuro, K.: Operations in open book foliations. Algebr. Geom. Topol. 14, 2983–3020 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Jankins, M., Neumann, W.: Lectures on Seifert Manifolds, Brandeis Lecture Notes, vol. 2. Brandeis University, Waltham (1983)

    Google Scholar 

  33. Kazez, W., Roberts, R.: Fractional Dehn twists in knot theory and contact topology. Algebr. Geom. Topol. 13(6), 3603–3637 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mitsumatsu, Y., Mori, A.: On Bennequin’s isotopy lemma, an appendix to convergence of contact structures to foliations. In: Walczak, P., et al. (eds.) Foliations 2005, pp. 365-371. World Sci. Publ., Hackensack (2006)

  35. Mosher, L.: Mapping class groups are automatic. Ann. Math. 142, 303–384 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pavelescu, E.: Braids and Open Book Decompositions. Ph.D. thesis, University of Pennsylvania (2008). http://www.math.upenn.edu/grad/dissertations/ElenaPavelescuThesis.pdf

  37. Pavelescu, E.: Braiding knots in contact 3-manifolds. Pac. J. Math. 253(2), 475–487 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Roussarie, R.: Plongements dans les variétés feuilletées et classification de feuilletages holonomie, I.H.E.S. Sci. Publ. Math. 43, 101–142 (1973)

    Article  MATH  Google Scholar 

  39. Thurston, W., Winkelnkemper, H.: On the existence of contact forms. Proc. Am. Math. Soc. 52, 345–347 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  40. Thurston, W.: Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle. Available at http://arxiv.org/pdf/math/9801045v1.pdf

  41. Thurston, W.: A norm on the homology of three-manifolds. Mem. Am. Math. Soc. 59, 99–130 (1986)

    MATH  Google Scholar 

  42. Rourke, C., Wiest, B.: Order automatic mapping class groups. Pac. J. Math 194(1), 209–227 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  43. Short, H., Wiest, B.: Ordering of mapping class groups after Thurston. Enseign. Math. 46, 279–312 (2000)

    MathSciNet  MATH  Google Scholar 

  44. Skora, R.: Closed braids in \(3\)-manifolds. Math. Z. 211, 173–187 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Ken Baker, John Etnyre, Charlie Frohman, William Kazez and Dale Rolfsen for helpful conversations, and Jesse Hamer and Sam Brensinger for helping with English. They especially thank the referee for numerous instructive comments and for pointing out gaps and typos. TI was partially supported by JSPS Postdoctoral Fellowships for Research Abroad. KK was partially supported by NSF Grants DMS-1016138 and DMS-1206770.

Author information

Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama Toyonaka, Osaka, 560-0043, Japan

    Tetsuya Ito

  2. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242, USA

    Keiko Kawamuro

Authors
  1. Tetsuya Ito
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Keiko Kawamuro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Keiko Kawamuro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ito, T., Kawamuro, K. Essential open book foliations and fractional Dehn twist coefficient. Geom Dedicata 187, 17–67 (2017). https://doi.org/10.1007/s10711-016-0188-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-016-0188-7

Keywords

Mathematics Subject Classification (1991)

Search

Navigation