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We provide a translation between Chekanov’s combinatorial theory for invariants of Legendrian knots in the standard contact R and a relative version of Eliashberg and Hofer’s contact homology. We use this translation to transport the idea of “coherent orientations” from the contact homology world to Chekanov’s combinatorial setting. As a result, we obtain a lifting of Chekanov’s differential graded algebra invariant to an algebra over Z[t, t] with a full Z grading.

Invariants of Legendrian Knots and Coherent Orientations