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Invariants of Legendrian knots and coherent orientations. (English) Zbl 1024.57014

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.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R17 Symplectic and contact topology in high or arbitrary dimension
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