Etnyre, John B.; Ng, Lenhard L.; Sabloff, Joshua M. Invariants of Legendrian knots and coherent orientations. (English) Zbl 1024.57014 J. Symplectic Geom. 1, No. 2, 321-367 (2002). A knot is Legendrian if it is everywhere tangent to the contact structure. Legendrian immersed curves in \(\mathbb{R}^3\) (with the standard contact structure) are naturally framed and the space of all Legendrian immersed curves has infinitely many connected components enumerated by the Maslov index \(r\) of a Legendrian curve. Legendrian knot theory studies examples of non-isotopic Legendrian knots that are isotopic as framed knots and have equal Maslov classes. The seminal constructions of the Differential Graded Algebra (DGA) associated to a Legendrian knot made by Chekanov, and of the contact homology made by Eliashberg and Hofer among many other things showed that Legendrian knot theory in \(\mathbb{R}^3\) is nontrivial. This interesting paper studies relations between DGA and relative contact homology. The authors use Floer’s and Hofer’s idea of coherent orientations to lift Chekanov’s DGA of a Legendrian knot with a Maslov class \(r\) from being an algebra with \(\mathbb{Z}_2\)-coefficients to being an algebra with \(\mathbb{Z}[t, t^{-1}]\)-coefficients and from being \(2r\)-graded to being \(\mathbb{Z}\)-graded. They show that the lifted DGA of Chekanov is a combinatorial translation of the relative contact homology of Eliashberg and Hofer. This result is extremely useful since it provides a much desired transition between the two theories and since, even though contact homology is much more general than the DGA, in many cases it is significantly harder to calculate. Reviewer: Vladimir Chernov (Hanover) Cited in 3 ReviewsCited in 29 Documents MSC: 57M27 Invariants of knots and \(3\)-manifolds (MSC2010) 57R17 Symplectic and contact topology in high or arbitrary dimension Keywords:contact homology; differential graded algebra; Legendrian knot; contact structure; Reeb field; Floer homology PDFBibTeX XMLCite \textit{J. B. Etnyre} et al., J. Symplectic Geom. 1, No. 2, 321--367 (2002; Zbl 1024.57014) Full Text: DOI arXiv