Modular forms and Dedekind symbols as topological invariants

首席研究员一直在研究Dedekind符号与结（流形）不变量之间的关系。例如，他在他的论文“Explicit formulae For two-bridge knot polynomial， J. Aust.”中证明了双桥结的Conway多项式是用Dedekind符号给出的。数学。社会法学，78(2005),149-166”。具有多项式互易律的戴德金符号尤为重要。他找到了具有多项式互易律的Dedekind符号的显式公式，并在论文《具有互易律的Dedekind符号，数学》中给出了结果。[j] .中文信息学报，2004,35(5):334 -334。众所周知，在Dedekind符号、模形式和周期多项式之间存在着天然的对应关系。他发现可以定义Dedekind符号上的Hecke算子，使它们与模形式和周期多项式上的Hecke算子兼容。作为推论，他得到了尖形赫克矩阵的显式公式。研究结果发表在论文《加权Dedekind符号上的Hecke算子》。数学。593(2006)1-29”和“顶点形式上的Hecke算子的显式公式，Dedekind符号和周期多项式，J. reine angew。”数学。（印刷）”。研究者Miyazawa研究了当结局部移动时结不变量是如何变化的。她在“结与低维流形”会议上发表了这一主题的演讲，并与Yasuhara合作发表了一篇论文“n分量Brunnian links up to C_n-move的分类，Topology, 153(2006) 1643-1650”。

The head investigator has been studying relationship between Dedekind symbols and knot (manifold) invariants. For example, he showed that Conway polynomials of two-bridge knot are given using Dedekind symbols in his paper "Explicit formulae for two-bridge knot polynomials, J. Aust. Math. Soc. 78 (2005), 149-166". Dedekind symbols with polynomial reciprocity laws are especially important. He found explicit formulas for Dedekind symbols with polynomial reciprocity laws and presented the result in his paper "Dedekind symbols with reciprocity laws, Math. Ann. 329 (2004), 315-334".It is known that there is natural correspondences between Dedekind symbols, modular forms and period polynomials. He found that Hecke operators on Dedekind symbols can be defined so that they are compatible with Hecke operators on modular forms and period polynomials. As a corollary, he obtained explicit formulas for Hecke matrices of cusp forms. The results are published in the papers "Hecke operators on weighted Dedekind symbols, J. reine angew. Math. 593 (2006) 1-29" and "Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials, J. reine angew. Math. (in print) ".The investigator Miyazawa studied how knot invariants change when knots are moved locally. She gave a talk on this subject at the meeting for "knots and low dimensional manifolds" and published a joint paper "Classification of n-component Brunnian links up to C_n-move, Topology Appl. 153 (2006) 1643-1650" with Akira Yasuhara.