Milnor numbers, Spanning Trees, and the Alexander-Conway Polynomial

الأصل · EN

We study relations between the Alexander-Conway polynomial ∇ₗ and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of ∇ₗ of an m-component link L all of whose Milnor numbers μᵢ₁... ᵢₚ vanish for p≤ n. We express this coefficient as a polynomial in Milnor numbers of L. Depending on whether the parity of n is odd or even, the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3-graphs into pairs of spanning trees. Our results complement determinantal formulas of Traldi and Levine obtained by geometric methods.