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
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.
Alexander-Conway polynomial, Milnor numbers, finite type invariants, Matrix-tree theorem, spanning trees, Pfaffian-tree polynomial
AMS subject classification.
Submitted to GT on 12 December 2001.
Paper accepted 22 July 2002.
Paper published 21 September 2002.
Notes on file formats
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