2002/08/27 by Joshua M. Sabloff, Sabloff, Joshua M. · 1 citation
Mathematics · #53D40 #57M27 #57R17 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.GT #math.SG #msc:53D40 #msc:57M27 #msc:57R17
paper · pdf · doi:10.48550/arxiv.math/0208214
49 pages, 39 figures
arxiv created 2002/08/27 · openalex publication_date 2002/08/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in E. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using ``Morse-Bott'' techniques. As an application, the invariant is used to distinguish two Legendrian knots that are smoothly isotopic, realize a non-trivial homology class, but are not Legendrian isotopic.