In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Legendrian knot often refers to a smooth embedding of the circle into which is tangent to the standard
contact structure
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ...
on It is the lowest-dimensional case of a
Legendrian submanifold, which is an embedding of a k-dimensional manifold into a
contact manifold
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution (differential geometry), distribution in the tangent bundle satisfying a condition called 'complete non-integrability' ...
that is always tangent to the contact hyperplane.
Two Legendrian knots are equivalent if they are isotopic through a family of Legendrian knots. There can be inequivalent Legendrian knots that are isotopic as topological knots. 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. This
Chekanov-Eliashberg invariant yields an invariant for loops of Legendrian knots by considering the monodromy of the loops. This has yielded noncontractible loops of Legendrian knots which are contractible in the space of all knots.
Any Legendrian knot may be
perturbed to a
transverse knot (a knot transverse to a contact structure) by pushing off in a direction transverse to the contact planes. The set of isomorphism classes of Legendrian knots modulo negative Legendrian stabilizations is in bijection with the set of transverse knots.
References
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External links
The Legendrian knot atlas
Knots and links
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