Geometry & Topology Monographs, Vol. 4 (2002),
Invariants of knots and 3-manifolds (Kyoto 2001),
Paper no. 13, pages 201--214.

Matrix-tree theorems and the Alexander-Conway polynomial

Gregor Masbaum

Abstract. This talk is a report on joint work with A. Vaintrob [arXiv:math.CO/0109104 and math.GT/0111102]. It is organised as follows. We begin by recalling how the classical Matrix-Tree Theorem relates two different expressions for the lowest degree coefficient of the Alexander-Conway polynomial of a link. We then state our formula for the lowest degree coefficient of an algebraically split link in terms of Milnor's triple linking numbers. We explain how this formula can be deduced from a determinantal expression due to Traldi and Levine by means of our Pfaffian Matrix-Tree Theorem [arXiv:math.CO/0109104]. We also discuss the approach via finite type invariants, which allowed us in [arXiv:math.GT/0111102] to obtain the same result directly from some properties of the Alexander-Conway weight system. This approach also gives similar results if all Milnor numbers up to a given order vanish.

Keywords. Alexander-Conway polynomial, Milnor numbers, finite type invariants, Matrix-tree theorem, spanning trees, Pfaffian-tree polynomial

AMS subject classification. Primary: 57M27. Secondary: 17B10.

E-print: arXiv:math.CO/0211063

Submitted to GT on 12 December 2001. Paper accepted 22 July 2002. Paper published 21 September 2002.

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Gregor Masbaum
Institut de Mathematiques de Jussieu, Universite Paris VII
Case 7012, 75251 Paris Cedex 05, France

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