Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Links in an open book decomposition and in the standard contact structure


Author: Hiroshi Matsuda
Journal: Proc. Amer. Math. Soc. 134 (2006), 3697-3702
MSC (2000): Primary 57M25, 53D10
Published electronically: June 8, 2006
MathSciNet review: 2240685
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study a relationship between arc presentations of links in $ \mathbb{R}^3$ and Legendrian links in $ \mathbb{R}^3$ with the standard tight contact structure. We determine the arc indices of torus knots.


References [Enhancements On Off] (What's this?)

  • 1. Daniel Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 87–161 (French). MR 753131
  • 2. Joan S. Birman and William W. Menasco, Special positions for essential tori in link complements, Topology 33 (1994), no. 3, 525–556. MR 1286930, 10.1016/0040-9383(94)90027-2
    Joan S. Birman and William W. Menasco, Erratum: “Special positions for essential tori in link complements [Topology 33 (1994), no. 3, 525–556; MR1286930 (95i:57006)], Topology 37 (1998), no. 1, 225. MR 1480888, 10.1016/S0040-9383(97)00027-X
  • 3. J. Birman, W. Menasco, Stabilization in the braid groups II: Transversal simplicity of knots, preprint available at arXiv:math.GT/0310280.
  • 4. Joan S. Birman and Nancy C. Wrinkle, On transversally simple knots, J. Differential Geom. 55 (2000), no. 2, 325–354. MR 1847313
  • 5. H. Brunn, Uber verknotete Kurven, in: Verhandlungen des ersten Internationalen Mathematiker-Kongresses, Zurich, 1897 (1898) 256-259.
  • 6. Peter R. Cromwell, Embedding knots and links in an open book. I. Basic properties, Topology Appl. 64 (1995), no. 1, 37–58. MR 1339757, 10.1016/0166-8641(94)00087-J
  • 7. I. Dynnikov, Arc-presentations of links. Monotonic simplification, preprint available at arXiv:math.GT/0208153.
  • 8. John B. Etnyre and Ko Honda, Knots and contact geometry. I. Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001), no. 1, 63–120. MR 1959579
  • 9. N. Kuhn, A conjectual inequality on the slice genus of links, Ph.D. Thesis, Princeton University, 1984.
  • 10. Herbert C. Lyon, Torus knots in the complements of links and surfaces, Michigan Math. J. 27 (1980), no. 1, 39–46. MR 555835
  • 11. Lee Rudolph, An obstruction to sliceness via contact geometry and “classical” gauge theory, Invent. Math. 119 (1995), no. 1, 155–163. MR 1309974, 10.1007/BF01245177

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57M25, 53D10

Retrieve articles in all journals with MSC (2000): 57M25, 53D10


Additional Information

Hiroshi Matsuda
Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, Hiroshima 739-8526, Japan
Address at time of publication: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
Email: matsuda@math.sci.hiroshima-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-06-08400-0
Keywords: Arc presentation, Legendrian knot
Received by editor(s): March 29, 2005
Received by editor(s) in revised form: June 28, 2005
Published electronically: June 8, 2006
Additional Notes: The author was partially supported by the Grant-in-Aid for Scientific Research (No.16740036), The Ministry of Education, Culture, Sports, Science and Technology, Japan.
Dedicated: Dedicated to Professor Yukio Matsumoto on his sixtieth birthday
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2006 American Mathematical Society