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On a negatively curved surface, we show that the Poincar{\'e} series countinggeodesic arcs orthogonal to some pair of closed geodesic curves has ameromorphic continuation to the whole complex plane. When both curves arehomologically trivial, we prove that the Poincar{\'e} series has an explicitrational value at 0 interpreting it in terms of linking number of Legendrianknots. In particular, for any pair of points on the surface, the lengths of allgeodesic arcs connecting the two points determine its genus, and, for any pairof homologically trivial closed geodesics, the lengths of all geodesic arcsorthogonal to both geodesics determine the linking number of the two geodesics.

Poincar{\'e} series and linking of Legendrian knots