Topological invariants of manifolds and modular forms

首席研究员一直在研究似乎是流形和纽结的拓扑不变量的数论函数。特别是，他一直在研究戴德金德的符号。他的研究是基于这样一个事实，即具有Laurent多项式互易定律的Dedekin符号双射地对应于模形式。他的第一个结果发表在《扭曲广义德德金符号，福原信治，J·努伯理论82(2000)47-78》一文中，这篇文章发展了扭曲德德金符号理论。第二个结果是发表在《非交换多项式互易定律，福原信治，松本幸雄，Noriko Yui，Internat，J.Math.12(2001)973-986》上的合著。证明了环面的拓扑与互易定律密切相关。第三个结果是找到了从雅可比形式获得Dedekind符号的方法。发表在《与J-形式及其互易律相关的代金符号》，Pukuhara Shji Pukuhara，J.数字理论98(2003)236-253。进一步，他发现了两桥结多项式的显式公式。这些公式是用新的代金德符号写成的。研究结果将发表在《澳大利亚人杂志》上。数学课。该项目的另一位研究者也得到了关于局部移动和Vassiliev不变量的结果，该结果的标题是《链环的Conway多项式的SCn-Moves和(n+L)-st系数》，宫泽春子Aida。

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".