STUDY ON KNOT INVARIANTS

点击翻译按钮获取中文摘要

We studied on topological invariants of knots and links, in particular, on Vassiliev invariants or finite type invariants. In a joint work with Shima and Habiro, we generalized these invariants to a higher dimension. A 2-knot in a 4-space is a ribbon 2-knot if it bounds an immersed 3-disk with only ribbon singularities. Making use of them, we gave a notion of finite type invariants for a class of ribbon 2-knots. Then we showed that each coefficient in the Taylor expansion of the normalized Alexander polynomial of a ribbon 2-knot is a Vassiliev invariant. Furthermore, Shima and Habiro have shown that the Alexander polynomial determines all the Vassiliev invariants of a ribbon 2-knot.Next, We gave an algorithm for calculating the second degree coefficient of the Conway polynomial of a ribbon 1-knot. This yields a recursive calculation for the Vassiliev invariant of order 2 for a ribbon 2-knot. This gives a partial answer to the question : Find a recursive formula for the Alexander polynomial for a higher dimensional knot. This problem is related to the question : Does there exist an invariant for a higher dimensional knot such as the Jones polynomial for a classical knot?We have been studying the basis of the space of Vassiliev invariants of lowere order for classical knots and links. As an application of this, we showed that some special values of the coefficient polynomials of the HOMFLY polynomial of a link are determined by the linking numbers. This generalizes a formula of Hoste and one of Lickorish and Millett.