$SC_n$-moves and the $(n+1)$-st Coefficients of the Conway Polynomials of Links

Abstract

A local move called a $C_n$-move is related to Vassiliev invariants. It is known that two knots are related by $C_n$-moves if and only if they have the same values of Vassiliev invariants of order less than $n$. In the link case, it is shown that a $C_n$-move does not change the values of any Vassiliev invariants of order less than $n$. It is also known that, if two links can be transformed into each other by a $C_n$-move, then the $n$-th coefficients of the Conway polynomials of them, which are Vassiliev invariants of order $n$, are congruent to each other modulo $2$. An $SC_n$-move is defined as a special $C_n$-move. It is shown that an $SC_n$-move does not change the values of any Vassiliev invariants of links of order less than $n+1$. In this paper, we consider the difference of the $(n+1)$-st coefficients of the Conway polynomials of two links which can be transformed into each other by an $SC_n$-move.