Legendrian and transversal knots in tight contact 3-manifolds.(English)Zbl 0809.53033

Goldberg, Lisa R. (ed.) et al., Topological methods in modern mathematics. Proceedings of a symposium in honor of John Milnor’s sixtieth birthday, held at the State University of New York at Stony Brook, USA, June 14-June 21, 1991. Houston, TX: Publish or Perish, Inc. 171-193 (1993).
Summary: Let $$(M, \xi)$$ be a contact 3-manifold, i.e., a 3-manifold with a completely nonintegrable tangent plane field $$\xi$$. Then there are two special types of knots in $$M$$ which are interesting to consider. The first one consists of Legendrian, i.e., tangent to $$\xi$$, knots and the second consists of knots transversal to $$\xi$$. Of course, it is natural to classify Legendrian and transversal knots up to Legendrian and transversal isotopy, respectively. This is the same as their classification up to the contact isotopy of the ambient manifold. In the present paper we discuss the status of the problem for tight contact manifolds and, in particular, for the standard contact sphere $$(S^ 3, \xi_ 0)$$.
For the entire collection see [Zbl 0780.00031].

MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 57M25 Knots and links in the $$3$$-sphere (MSC2010)