No Access

LEGENDRIAN KNOTS AND LINKS CLASSIFIED BY CLASSICAL INVARIANTS

FAN DING

Department of Mathematics, Peking University, Beijing 100871, P. R. China

Search for more papers by this author

and

HANSJÖRG GEIGES

Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln, Germany

Search for more papers by this author

https://doi.org/10.1142/S0219199707002381Cited by:16 (Source: Crossref)

Abstract

It is shown that Legendrian (respectively transverse) cable links in S³ with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston–Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J¹(S¹) with its standard tight contact structure.

Keywords:

AMSC: 53D35, 57M25

References

G. Burde and H. Zieschang , Knots , de Gruyter Stud. Math. 5 ( Walter de Gruyter & Co. , Berlin , 1985 ) . Google Scholar
Yu. Chekanov, Invent. Math. 150, 441 (2002). Crossref, Web of Science, Google Scholar
F. Ding and H. Geiges, A Legendrian surgery presentation of contact 3-manifolds, Math. Proc. Cambridge Philos. Soc.136 (2004) pp. 583–598. Google Scholar
Ya. Eliashberg, Ann. Inst. Fourier (Grenoble) 42, 165 (1992). Crossref, Web of Science, Google Scholar
Ya. Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds, Topological Methods in Modern Mathematics (Publish or Perish, Houston, 1993) pp. 171–193. Google Scholar
M. Fraser Ya. Eliashberg, Geometry, Topology, and Dynamics (Montréal, 1995), CRM Proc. Lecture Notes 15 (American Mathematical Society, Providence, 1998) pp. 17–51. Google Scholar
J. B. Etnyre, Topology and Geometry of Manifolds (Athens, GA, 2001), Proc. Sympos. Pure Math. 71 (American Mathematical Society, Providence, 2003) pp. 81–107. Crossref, Google Scholar
J. B. Etnyre, Handbook of Knot Theory (Elsevier, Amsterdam, 2005) pp. 105–185. Crossref, Google Scholar
J. B. Etnyre and K. Honda, J. Symplectic Geom. 1, 63 (2001). Crossref, Web of Science, Google Scholar
H. Geiges, Handbook of Differential Geometry 2 (North-Holland, Amsterdam, 2006) pp. 315–382. Crossref, Google Scholar
E. Giroux, Comment. Math. Helv. 66, 637 (1991). Crossref, Web of Science, Google Scholar
E. Giroux, Invent. Math. 141, 615 (2000). Crossref, Web of Science, Google Scholar
E. Giroux, Sur les transformations de contact au-dessus des surfaces, Essays on Geometry and Related Topics2, Monogr. Enseign. Math., No.38, eds. É. Ghyset al. (Genève, 2001) pp. 329–350. Google Scholar
K. Honda, Geom. Topol. 4, 309 (2000). Crossref, Web of Science, Google Scholar
Y. Kanda, Invent. Math. 133, 227 (1998). Crossref, Web of Science, Google Scholar
J. Milnor , Singular Points of Complex Hypersurfaces , Annals of Mathematics Studies 61 ( Princeton University Press , 1968 ) . Google Scholar
L. Ng, Topology 42, 55 (2003). Crossref, Web of Science, Google Scholar
L. Ng and L. Traynor, J. Symplectic Geom. 2, 411 (2004). Crossref, Web of Science, Google Scholar
S. Smale, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc.10 (1959) pp. 621–626. Google Scholar