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The historical significance of Herbert Seifert's paper 'Ueber das Geschlecht von Knoten'
by
Mark E. Kidwell
U.S. Naval Academy
In one remarkable paper, Seifert gave a combinatorial algorithm for spanning an orientable, singularity-free surface in a knot in the 3-sphere, defined a matrix that describes the homology relations of this embedded surface, displayed a new and efficient method of computing the Alexander polynomial from this "Seifert matrix", and showed that half the degree of the Alexander polynomial gives a lower bound for the genus of the "Seifert surface". He also used his matrix to demonstrate exactly which polynomials can be Alexander polynomials of knots, a feat as yet unduplicated for any of the post-Jones polynomials. He provided an example of an apparently knotted curve that has Alexander polynomial 1 and, in a final flourish, proved its knottedness using hyperbolic geometry. If time permits, we will give a hopelessly incomplete review of further developments that stemmed from Seifert's brilliant paper.
Date received: November 29, 2007
Copyright © 2007 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cavo-12.