In Penne, 1995, we observed that the multi-variable Alexander polynomial can be computed by means of "multi-variable Burau matrices", the entries of which can be visualized by a "path model". In this article we introduce the Alexander polynomial for line configurations. The correspondence between "adjacency of lines" in a configuration, and the factorization of its Alexander polynomial can be well understood by way of the path model.
Introduction.In a previous paper ([12]) we introduced a new machinery to compute the multi-variable Alexander polynomial ∆ L of a link L with k components. Although everything was stated for pure links, the techniques of [12] apply for general links if one makes a small additional observation (Theorem 1).Let us quickly repeat the basic facts of the (non-reduced) Alexander polynomial. If we regard the link L as a closed braid β, β ∈ B n , and if we use Artin's representation of the braid group as free automorphisms,we can present the group of L, G(L) (the fundamental group of its complement), by n generators x 1 , . . . , x n subject to the relationsFor more details, we refer to [2,3]. It is well known that each of the n relations in (2) is implied by the others, allowing us to drop one relation, and we come up with a group presentation for G(L) of "deficiency 1" (one less relation than generators): , and extend it to the corresponding group rings, it can be applied to the entries of the Jacobian, resulting in an Alexander matrix A(L) for L. We refer to [8] where a : Z[F n ] → Z[H] composes the quotient map with the abelianizer. Notice thatWe conclude that the Alexander polynomial of a link L = β is immediately obtained once we have computed A(β). In this article, A(β) will be computed by associating so-called multi-variable Burau matrices with the crossings of β, multiplying them in order of appearance. This generalizes the well-known procedure for knots (k = 1), where A(β) is obtained as the Burau representation τ B (β) of β.The fundamental properties of the multi-variable Burau matrices can be found in [12]. In this article we focus on some of their applications, showing how they can improve our understanding of the Alexander polynomial. As pointed out in [12], the entries of a multi-variable Burau matrix can be easily visualized by considering topological paths on the involved braid diagram, which are evaluated by appropriate weights. This path model will be explained in Section 4. The main goal of this article is to illustrate the elegance of the path model by obtaining fundamental combinatorial proofs for classical theorems of Morton, Torres and Sumners-Woods on the Alexander polynomial (Section 5 and Section 6).
THE ALEXANDER POLYNOMIAL 317Most of our material will be stated in terms of a special class of links. More precisely, we are motivated by the isotopy problem of line configurations, that are configurations of a finite number of lines in RP 3 . Two line configurations are called rigidly isotopic if there exists an ambient isotopy that connects them and during which the ...