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Alexander-Conway polynomial, milnor numbers, and the Pfaffian matrix-tree theorem

Thursday 10th April 2008 - 11:30 to 12:30
INI Seminar Room 1

The well-know classical theorem of Kirchhoff expresses the spanning-tree polynomial of a graph as a determinant. Motivated by questions from knot theory, A. Vaintrob and I discovered in 2000 a new matrix-tree theorem which expresses the spanning-tree polynomial of a 3-uniform hypergraph as a Pfaffian. Other proofs of our theorem have been given by Abdesselam and Hirschman-Reiner, and generalisations have been obtained by Abdesselam and Caracciolo-Sokal-Sportiello.

In my talk, I plan to go back to the origins and explain how both the classical matrix-tree theorem and our Pfaffian-tree theorem appear naturally in knot theory when studying the Alexander-Conway polynomial of a link from the point of view of Feynman diagrams. This approach leads to generalizations of our formula involving higher order Milnor numbers. The talk will however be elementary and no previous knowledge of knot theory will be assumed.

Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons