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Matrix-tree theorems and the Alexander–Conway polynomial

Gregor Masbaum

Geometry & Topology Monographs 4 (2002) 201–214

DOI: 10.2140/gtm.2002.4.201


This talk is a report on joint work with A Vaintrob [arXiv:math.CO/0109104] and [arXiv: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.


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

Mathematical Subject Classification

Primary: 57M27

Secondary: 17B10


Received: 12 December 2001
Accepted: 22 July 2002
Published: 21 September 2002

Gregor Masbaum
Institut de Mathématiques de Jussieu
Université Paris VII
Case 7012
75251 Paris Cedex 05