Legendrian and transversal knots. (English) Zbl 1095.57006

Etnyre, John B.

Menasco, William (ed.) et al., Handbook of knot theory. Amsterdam: Elsevier (ISBN 0-444-51452-X/hbk). 105-185 (2005).

This survey is an introduction to the study of Legendrian knots and an overview of the current state of knot theory in contact geometry. The author concentrates on Legendrian and transversal knots in dimension three where their theory is most fully developed and where they are most intimately tied to topology (in this dimension we can use a predominantly topological and combinatorial approach to their study). Some of the problems presented in this survey are: Invariants of Legendrian and transversal knots, Tightness and bounds on invariants, contact homology, decomposition invariants, classification results of Legendrian knots, Legendrian Knots in \(\mathbb R^{2n+1}\), Generalisation of the Chekanov-Eliashberg DGA, Applications: Legendrian surgery, Knot concordance, invariants of classical knots, contact homology and topology and topology, knot invariants. At the end an extensive bibliography is given.
For the entire collection see [Zbl 1073.57001].

Reviewer: Corina Mohorianu (Iaşi)

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
57R17	Symplectic and contact topology in high or arbitrary dimension
57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes

Keywords:

knots; links; abstract topology; cables; invariants of classical knots