Poincar{é}级数与Legendrian纽结的连接
Poincar{é} series and linking of Legendrian knots
摘要：
在负曲面上，我们证明了与某些闭测地线曲线对正交的测地弧的Poincar级数对整个复平面具有亚纯延拓.当这两条曲线是同调平凡的，我们证明了庞加莱{é}系列有一个明确的合理的值在0解释它的环绕数的勒让德结。特别地，对于曲面上的任何一对点，连接这两点的所有测地弧的长度决定了它的亏格，而对于任何一对同调平凡的闭测地线，与这两条测地线正交的所有测地弧的长度决定了这两条测地线的环绕数。
英文摘要：
On a negatively curved surface, we show that the Poincar{é} series counting geodesic arcs orthogonal to some pair of closed geodesic curves has a meromorphic continuation to the whole complex plane. When both curves are homologically trivial, we prove that the Poincar{é} series has an explicit rational value at 0 interpreting it in terms of linking number of Legendrian knots. In particular, for any pair of points on the surface, the lengths of all geodesic arcs connecting the two points determine its genus, and, for any pair of homologically trivial closed geodesics, the lengths of all geodesic arcs orthogonal to both geodesics determine the linking number of the two geodesics.
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