Legendrian knots are imbeddings of a circle in a contact 3-manifold which istangent to the contact plane at every point. For certain standard compact 3-manifolds, invariants of Legendrian knots other than the ordinary knot type are available. Usually, one my have “index-type” invariants which characterize the homotopy types of Legendrian curves. One may also have a “self-linking number” coming from the fact that a Legendrian knot admits a natural framing. So the key question in the study of Legendrian knots seems to be whether two Legendrian knots with the same ordinary framed knot type and the same “indices” are necessarily Legendrian isotopic.See.This question is still open and the purpose of this article is to offer a new approach to this question. This new approach leads to a combinatorial question (Question 7.1) of the same natural as the geometric question about Legendrian knots. It is very likely that these two questions, the geometric one and the combinatorial one, are in fact equivalent. We provide also some evidence supporting a negative answer to the combinatorial question.
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