Topological invariants of manifolds and modular forms

首席研究员一直在研究数论函数，这些函数似乎是流形和结点的拓扑不变量。特别是他一直在研究戴德金符号。他的研究是基于这样一个事实，即具有洛朗多项式互易律的戴德金符号与模形式双射对应。他的第一个结果是发表在文章“扭曲的广义Dedekind符号，Shinji Fukuhara，J. Nunber Theory 82（2000）47-78”，其中开发了扭曲的Dedekind符号的理论。第二个结果是一个联合工作，发表在“非交换多项式互易律，Shinji Fukuhara，Yukio松本，Noriko Yui，Internat. 12（2001）973-986”。证明了环面的拓扑结构与互易律密切相关。第三个结果是找到了从Jacobi形式得到Dedekind符号的方法。它发表在“与J-形式及其互反律相关的Dedekind符号，Shinji Pukuhara，J.数论98（2003）236-253”中。进一步，他发现了两桥纽结多项式的显式公式。这些公式用新的戴德金符号表示。该结果将发表在J. Austral. Math. Soc上。该项目的其他研究人员也获得了关于局部移动和Vassiliev不变量的结果，该结果准备以“链接的Conway多项式的SCn-移动和（n +1）-st系数”为标题发表，Haruko Aida Miyazawa。

The head investigator has been studying the number theoretical functions which appear to be topological invariants of manifolds and knots. In particular he has been studying Dedekind symbols. His research is based on the fact that Dedekind symbols with Laurent polynomial reciprocity law correspond to modular forms bijectively. His first result is published in the article "Twisted generalized Dedekind symbols, Shinji Fukuhara, J. Nunber Theory 82 (2000) 47-78" which developed the theory of twisted Dedekind symbols.The second result is a joint work which was published in "Non-commutative polynomial reciprocity law, Shinji Fukuhara, Yukio Matsumoto, Noriko Yui, Internat. J. Math. 12 (2001) 973-986". Here the authors demonstrate that topology of a torus is closely related to reciprocity law.The third result is finding the method of obtaining Dedekind symbols from Jacobi forms. It was published in "Dedekind symbols associated with J-forms and their reciprocity law, Shinji Pukuhara, J. Number Theory 98 (2003) 236-253". Futher- ***re he discovered the explicit formulae for two-bridge knot polynomials. The formulae are written in terms of new Dedekind symbols. The result will be published in J. Austral. Math. Soc.The other investigator of this project also obtained the result on local moves and Vassiliev invariants which is prepared to be published with the title "SCn-moves and (n + l)-st coefficients of the Conway polynomials of links, Haruko Aida Miyazawa".