Studies of local moves and finite type invariants in knot theory

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In 1990, Vassiliev invariants for knots were defined. They order all knot invariants and they are also called finite type invariants. The first aim of this research is to study the finite type invariants by combinatorial methods. The start point is the following result proved by Goussarov and Habiro independently ; two knots have the same Vassiliev invariants of order less than n if and only if they can be transformed into each other by a finite sequence of C_n-moves.As a joint work with Prof. Yasutaka Nakanishi, we have that for any given pair of a natural number n and a knot K, there exist infinitely many knots whose Vassiliev invariants of order less than or equal to n and Conway polynomials coincide with those of K. In the finite type invariants, the coefficients of the Conway polynomial are not powerful to classify the knots.C_n-moves may change the Vassiliev invariants of order n. As a joint work with Harumi Yamada, we showed that a standard C_n-move can change the coefficient of z^n by 0 or ±2. It is possible to say that we nearly cleared the relation between C_n-moves and the coefficients of the Conway polynomial.We can define the simplicial complex for the set of knots by using C_n-moves and it is called the C_n-Gordian complex of knots. Let K be a knot and K^<C_n> the set of knots obtained from K by a single C_n-move. We showed that there are knots K_1 and K_2 such that they have the same Conway polynomial and the sets of Conway polynomials of K_1^<C_n> and those of K_2^<C_n> do not coincide, as a joint work with Prof. Yasutaka Nakanishi. This theorem are related to the C_n-Gordian complex and the Conway polynomial and we consider an expansion of the result.