Matrix-tree theorems and the Alexander-Conway polynomial

Details Matrix-tree theorems and the Alexander-Conway polynomial Matrix-tree theorems and the Alexander-Conway polynomial

2002-11-04

arxiv.org/abs/math/0211063v1

Geom. Topol. Monogr. 4 (2002) 201-214

Mathematics

Combinatorics

Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper13.abs.html

Scientific paper

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.

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