Invariants of Legendrian Knots and Coherent Orientations

Abstract

We provide a translation between Chekanov's combinatorial theory for invariants of Legendrian knots in the standard contact R 3 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 −1 ] with a full Z grading. JBE is partially supported by an NSF Postdoctoral Fellowship (Grant # DMS-0072853). LLN is partially supported by grants from the NSF and DOE. JMS is partially supported by an NSF Graduate Student Fellowship and an ARCS Fellowship.

Rational symplectic field theory for Legendrian knots

“…There are, however, two caveats to this identification. First, the signs used here do not coincide precisely with the signs from [ENS02], though they do agree with another sign assignment for Legendrian contact homology given in [EES05b]. However, up to a basis change, all possible sign assignments are equivalent.…”

“…[Che02,ENS02] for the precise definition). Note that on the quotient level, any basis change fixes t and t −1 .…”

“…Thus the differential d on comm is independent of •, and only depends on * insofar as * keeps track of powers of t in the SFT differential, cf. group-ring coefficients in Legendrian contact homology [ENS02].…”

New Invariants of Legendrian Knots

“…This DGA is related to the symplectic field theory introduced by Eliashberg, Givental, and Hofer in [7] (see [10]). …”

“…However, the construction described above can be modified to associate with a Legendrian knot L a DGA graded by Z and having Z[s, s −1 ] (where deg(s) = m(L)) as a coefficient ring [10]. After reducing the grading to Z/m(L)Z, and applying the homomorphism Z[s, s −1 ] → Z/2Z sending both s and 1 ∈ Z to 1 ∈ Z/2Z, this Z[s, s −1 ]-DGA becomes the Z/2Z-DGA of the knot L.…”