In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into
It is the lowest-dimensional case of a Legendrian submanifold, which is an embedding of a k-dimensional manifold into a (2k+1)-dimensional contact manifold that is always tangent to the contact hyperplane.
Many inequivalent Legendrian knots can be distinguished by considering their Thurston-Bennequin invariants and rotation number, which are together known as the "classical invariants" of Legendrian knots.
More sophisticated invariants have been constructed, including one constructed combinatorially by Chekanov and using holomorphic discs by Eliashberg.
The set of isomorphism classes of Legendrian knots modulo negative Legendrian stabilizations is in bijection with the set of transverse knots.