On the Alexander polynomials of cobordant links

Abstract

R.H. Fox and J.W. Milnor in [4] showed that the Alexander polynomial of a slice knot is of the form f(f)f(t~) for an integral polynomial f ( t ) with I y( 1)|=1. This clearly implies that the Alexander polynomials of cobordant knots are identical up to the polynomials of the form f(i)f(t~). The purpose of this paper is to generalize this property to that of arbitrary cobordant links. On the basis of the work done by K. Reidemeister, H.G. Shumann and W. Burau, R.H. Fox defined the /^-variable Alexander polynomial A°(tly •••, tμ) of a link L with μ components, (cf. R.H. Fox [3], G. Torres [9].) One difficulty in our study is that using this definition the polynomial A°(tly •• ,/μ) vanishes for many links. For example, any decomposable link (that is, a link separated into two sublinks by a 2-sρhere within a 3-sphere) has A°(tly •••, tμ)=0. To avoid this difficulty we shall re-define the Alexander polynomial A(ΐly •••, tμ) so that it is always a non-zero polynomial. To measure the difference between AQ(ΪI, •••, tμ) and A(tly •••, ίμ), we will also introduce a numerical invariant β(L ) with 0</3(L)<μ-l such that