MODULAR COCYCLES AND LINKING NUMBERS
MODULAR COCYCLES AND LINKING NUMBERS
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DOI：
10.1215/00127094-3793032
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发表时间：
2017-04-15
影响因子：
2.5
通讯作者：
Toth, A.
中科院分区：
文献类型：
--
作者：
Duke, W.;Imamoglu, O.;Toth, A.
已知3流形SL(2， Z) \ SL（2， R）与S-3中三叶结的补是微分同构的。E. Ghys证明了这种三叶结与模结的连接数由Rademacher符号给出，该符号是经典Dedekind符号的一种均质化。Dedekind符号是历史上在SL（2， Z）下Dedekind的eta函数的对数变换公式中出现的。本文给出了与固定模结相关的Dedekind符号的推广。这个符号也出现在某个模函数的变换公式中。它可以用狄利克雷级数的一个特殊值来计算，并且满足互易律。该符号的均质化推广了Rademacher符号，给出了由模结组成的两个不同对称链路之间的连接数。
It is known that the 3-manifold SL(2, Z) \ SL(2, R) is diffeomorphic to the complement of the trefoil knot in S-3. E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind's eta function under SL(2, Z). In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It can be computed in terms of a special value of a certain Dirichlet series and satisfies a reciprocity law. The homogenization of this symbol, which generalizes the Rademacher symbol, gives the linking number between two distinct symmetric links formed from modular knots.