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

Gregor Masbaum·Arkady Vaintrob
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Abstract

We study relations between the Alexander-Conway polynomial $\nabla_L$ 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 $\nabla_L$ of an m-component link L all of whose Milnor numbers $μ_{i_1... i_p}$ vanish for $p\le n$. 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.

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Gregor Masbaum, Arkady Vaintrob. 2001-11-08. Milnor numbers, Spanning Trees, and the Alexander-Conway Polynomial. https://arxiv.org/abs/math/0111102

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