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

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Gregor Masbaum, Arkady Vaintrob

Geometric Topology · Submitted 8 Nov 2001

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We give a formula for the first non-vanishing coefficient of $\nabla_L$ of an m-component link L all of whose Milnor numbers $\mu_{i_1... i_p}$ vanish for $p\le n$.

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arXiv IDmath/0111102 (version 1)
Pith IDMWDFDCDE, Pith's permanent record of this paper and its review
CitationArkady Vaintrob, Gregor Masbaum. "Milnor numbers, Spanning Trees, and the Alexander-Conway Polynomial." arXiv:math/0111102 (2001). https://pith.science/paper/MWDFDCDE
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arXiv listing abstract v1. Results and the main claim are underlined; .

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 $\mu_{i_1... i_p}$ 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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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. contribution 0.80 0.79 0.10 asserted 0.83
s1 We give a formula for the first non-vanishing coefficient of $\nabla_L$ of an m-component link L all of whose Milnor numbers $\mu_{i_1... i_p}$ vanish for $p\le n$. result 0.70 0.95 0.35 asserted 1.00
s2 We express this coefficient as a polynomial in Milnor numbers of L. result 0.83 0.95 0.22 asserted 1.00
s3 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. result 0.92 0.91 0.34 asserted 1.00
s4 Our results complement determinantal formulas of Traldi and Levine obtained by geometric methods. interpretation 0.70 0.38 0.06 asserted 1.00

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Original 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 $\mu_{i_1... i_p}$ 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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