Legendrian and Transversal Knots

Contact structures on manifolds and Legendrian and transversal knots in them are very natural objects, born over two centuries ago, in the work of Huygens, Hamilton and Jacobi on geometric optics and work of Lie on partial differential equations. They touch on diverse areas of mathematics and physics, and have deep connections with topology and dynamics in low dimensions. The study of Legendrian knots is now a rich and beautiful theory with many applications. This survey is an introduction to, and overview of the current state of knot theory in contact geometry. For a discussion of the driving questions in the field see [38] and for a more historical discussion of contact geometry see [48]. This paper will concentrate on Legendrian and transversal knots in dimension three where their theory is most fully developed and where they are most intimately tied to topology. Moreover, in this dimension one may use a predominately topological and combinatorial approach to their study. In Section 6 we will make our only excursion into the study of higher dimensional Legendrian knots. Throughout this survey we assume the reader is familiar with basic topology at the level of [76]. We have tried to keep the contact geometry prerequisites to a minimum, but it would certainly be helpful to have had some prior exposure to the basics as can be found in [33, 49]. Some of the proofs in Section 5 rely on convex surface theory which can be found in [33], but references to convex surfaces can be largely ignored without serious loss of continuity.