论文标题
链接模块结的数量
Linking numbers of modular knots
论文作者
论文摘要
模块化的PSL（2; Z）用模块化表面M的商在双曲线平面HP上作用，其单位切线束U是3个球体中Trefoil结的补充。 PSL（2; Z）的双曲线结合类别对应于M中的封闭式的测量学，它们升至u中的测量流的周期性轨道，该轨道定义了模块化结。模块结和三叶线之间的链接数是充分理解的。确实，埃蒂安·吉斯（Etienne Ghys）在2006年表明，它们是由相应共轭类别的Rademacher不变的。 Rademacher函数是PSL（2; Z）的同质准态，他在1992年与Jean Barge认识到它是有界Euler类的原始性的一半。这阐明了迈克尔·阿蒂亚（Michael Atiyah）关于Dedekind ETA功能的对数的1987年作品，该函数以不少于数学领域中出现的其他六个重要功能确定了它。我们关注模块化结之间的链接数，并通过算术，组合，拓扑和组理论口味得出多个公式。特别是，我们将其与一对模块化结相关联，在PSL（2; Z）的字符品种上定义的函数，其边界点的极限恢复了它们的链接数。此外，我们表明与模块化结的链接数减去，该链接及其反向在模块化组上产生均匀的准态度，以及如何从中提取自由基础。为此，我们证明链接配对是非退化的。
The modular group PSL(2;Z) acts on the hyperbolic plane HP with quotient the modular surface M, whose unit tangent bundle U is a 3-manifold homeomorphic to the complement of the trefoil knot in the 3-sphere. The hyperbolic conjugacy classes of PSL(2;Z) correspond to the closed oriented geodesics in M. Those lift to the periodic orbits for the geodesic flow in U, which define the modular knots. The linking numbers between modular knots and the trefoil is well understood. Indeed, Etienne Ghys showed in 2006 that they are given by the Rademacher invariant of the corresponding conjugacy classes. The Rademacher function is a homogeneous quasi-morphism of PSL(2;Z) which he had recognised with Jean Barge in 1992 as half the primitive of the bounded euler class. This shed light on the 1987 work of Michael Atiyah concerning the logarithm of the Dedekind eta function which identified it with no less than that six other important functions appearing in diverse areas of mathematics. We are concerned with the linking numbers between modular knots and derive several formulae with arithmetical, combinatorial, topological and group theoretical flavours. In particular we associate to a pair of modular knots a function defined on the character variety of PSL(2;Z), whose limit at the boundary point recovers their linking number. Moreover, we show that the linking number with a modular knot minus that with its inverse yields a homogeneous quasi-morphism on the modular group, and how to extract a free basis out of these. For this we prove that the linking pairing is non degenerate.
