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Alexander-Conway Polynomial, Milnor numbers, and a new Matrix-Tree Theorem
Geometry/Topology| Speaker: | Gregor Masbaum, Institut de Math |
| Location: | 693 Kerr |
| Start time: | Tue, Feb 12 2002, 2:10PM |
Description
The lowest degree coefficient of the Alexander-Conway
polynomial of an algebraically split link can be expressed via
Milnor's triple linking numbers in two different ways. One way is
via a determinantal expression due to Levine. Using the
Alexander-Conway weight system, we give another expression in
terms of spanning trees on a 3-graph. The equivalence of the
two answers is explained by a new matrix-tree theorem, relating
enumeration of spanning trees in a 3-graph and the Pfaffian of a
certain skew-symmetric matrix associated with it. Similar results
for the lowest degree coefficient of the Alexander-Conway
polynomial exist if all Milnor numbers up to a given order vanish.
(Joint work with A. Vaintrob)