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We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in    P  ×  R   , where   P   is a punctured Riemann surface. As an application we show that, for any integer   k   and any homology class    h  ∈   H  1    (  P  ×  R  )    , there are   k   Legendrian knots, all representing   h  , which are pairwise smoothly isotopic through a formal Legendrian isotopy, but which lie in mutually distinct Legendrian isotopy classes.

Legendrian contact homology in the product of a punctured Riemann surface and the real line