#### 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.

Notes on file formats
Gregor Masbaum

Institut de Mathematiques de Jussieu, Universite Paris VII

Case 7012, 75251 Paris Cedex 05, France

Email: masbaum@math.jussieu.fr

GTM home page

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