ALEXANDER POLYNOMIAL FOR LINK CROSSINGS

We define a crossing of a link without referring to a specific projection of the link and describe a construction of a non-normalized Alexander polynomial associated to collections of such crossings of oriented links under an equivalence relation, called homology relation. The polynomial is computed from a special Seifert surface of the link. We prove that the polynomial is well-defined for the homology equivalence classes, investigate its relationship with the combinatorially defined Alexander polynomials and study some of its properties.