Legendrian knots are smooth knots whose tangent directions are contained in a contact structure , such as the standard contact structure on $\mathbb{R}^3$, $dz=y~dx$. Every knot has Legendrian representatives. Two Legendrian knots of the same topological type might not be isotopic through Legendrian knots.
The projection of a Legendrian knot in the standard contact structure to the $xz$ plane if xz$-plane is called its front projection. Some people study Legendrian knots through diagrams showing front projections. The$y$coordinate can be recovered from the slope, so all crossings are determined by the diagram. However, there can be no vertical tangencies, since$y$would be undefined, and you must allow cusps. See this Notices article. There are analogues of Reidemeister moves, so this gives a refinement of knot theory described by a set of diagram moves on front projections. Actually, you don't have to work with cusped diagrams. You can make all cusps horizontal, and you could choose to replace the horizontal cusps with vertical tangencies. So, standard knot diagrams with up to a restricted set of moves (including disallowing some isotopies where no Reidemeister move was performed, but where vertical tangencies would have been introduced or removed) are equivalent to Legendrian knots. Link Isotopyisotopy Suppose you study curves up to isotopy instead of the standard ambient isotopy used in knot theory. You may be disappointed: knot theory in$S^3$becomes trivial. You are allowed to replace a piece of a diagram showing a long knot with a long unknot by shrinking the diagram knot to a point and forgetting it. However, link theory is still nontrivial, and so is knot theory in a$3$-manifold which is not simply connected. See Rolfsen, "Localized Alexander Invariants and Isotopy of LinksLinks." Annals of Mathematics 101 (1975) 1-19. 1 Legendrian knots Legendrian knots are smooth knots whose tangent directions are contained in a contact structure, such as the standard contact structure on$\mathbb{R}^3$,$dz=y~dx$. Every knot has Legendrian representatives. Two Legendrian knots of the same topological type might not be isotopic through Legendrian knots. The projection of a Legendrian knot in the standard contact structure to the$xz$plane if called its front projection. Some people study Legendrian knots through diagrams showing front projections. The$y$coordinate can be recovered from the slope, so all crossings are determined by the diagram. However, there can be no vertical tangencies, since$y$would be undefined, and you must allow cusps. See this Notices article. There are analogues of Reidemeister moves, so this gives a refinement of knot theory described by a set of diagram moves on front projections. Actually, you don't have to work with cusped diagrams. You can make all cusps horizontal, and you could choose to replace the horizontal cusps with vertical tangencies. So, standard knot diagrams with a restricted set of moves (including disallowing some isotopies where no Reidemeister move was performed, but where vertical tangencies would have been introduced or removed) are equivalent to Legendrian knots. Link Isotopy Suppose you study curves up to isotopy instead of the standard ambient isotopy used in knot theory. You may be disappointed: knot theory in$S^3$becomes trivial. You are allowed to replace a piece of a diagram showing a long knot with a long unknot by shrinking the diagram to a point and forgetting it. However, link theory is still nontrivial, and so is knot theory in a$3\$-manifold which is not simply connected. See Rolfsen "Localized Alexander Invariants and Isotopy of Links" Annals of Mathematics 101 (1975) 1-19.