Abstract
Contact structures on manifolds and Legendrian and transversal knots in them are very natural objects, born over a century ago, in the work of Huygens, Hamilton, and Jacobi on geometric optics and work of Lie on partial differential equations. This chapter discusses 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 a predominately topological and combinatorial approach may be used to their study. The chapter then examines higher dimensional Legendrian knots. The chapter also explains the concept of Lagrangian protection. Analogous to the definition of finite type invariants of topological knots, finite type invariants of Legendrian knots can be defined. It is then possible to get a global contact isotopy if there is a fixed over twisted disk in the complement of both Legendrian knot. A complete understanding of Legendrian unknots in over twisted contact structures must wait for future work.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
