Modular forms and Dedekind symbols as topological invariants

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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.