The Alexander polynomial in several variables can be expressed in terms of the Vassiliev invariants

Abstract

The definition of an invariant of finite order for links devised by Vassiliev [1] has led to the construction of a beautiful theory that includes almost all the known polynomial invariants. Bar-Natan [2] and Birman and Lin [3] proved that if two links have all their Vassiliev invariants equal, then their HOMFLY and Kauffman polynomials are also equal, and, as special cases of HOMFLY, so are the Jones and Conway polynomials and the Alexander polynomial in one variable. The Alexander polynomial in several variables (whose description in [4] by means of local relations is substantially more complicated) is missing from this list. The present note fills this gap. The Alexander polynomial in several variables is defined for ordered oriented links [5]. Each component has its own corresponding variable. For ordered links the definition of an invariant of finite order needs no modification. In the standard definition of the Alexander polynomial there is a lack of uniqueness in the choice of sign and in multiplication by a monomial of the form tj. This is related to the fact that the Alexander polynomial is an invariant of torsion type (see [6]) and creates difficulties in the study of its interrelations with invariants of finite order. We shall give below a definition of a normalized Alexander polynomial in several variables, which is uniquely defined by a diagram, and we shall show that after a certain change of variables all its coefficients are Vassiliev invariants. We shall give the definition straight away for links in R3 that are allowed to have finitely many points of self-intersection. We shall consider links with at least two components. We begin the definition of the normalized Alexander polynomial with a simple algebraic assertion. First let us introduce the following notation. For a matrix A = (dij)i^ij^n we denote by A = (2ij)i^ij Δ^(Λ), where Aji(A) is the minor complementary to a,;.