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Signature Jumps and Alexander Polynomials for Links
by
Patrick M. Gilmer
Louisiana State University
Coauthors: Charles Livingston
Given a spanning surface for an oriented link, one may associate a family of Hermitian forms which are parametrized by the points on the unit circle in the complex plane. The signatures of these forms do not depend on the choice of a spanning surface, and thus define an integer valued function on the unit circle. It is a step function. The spanning surface can also be used to define a sequence of polynomials called the Alexander polynomial and higher Alexander polynomials of the link.
A link with one component is a knot. For knots, it is well-known that the signature function can only jump at roots of the Alexander
polynomial and the jump at a root is less or equal to the multiplicity of the root.
What can one say if the link has more than one component? Then it may happen that the Alexander polynomial is identically zero. It turns out that a similar result holds about the location and size of the jumps if one replaces the Alexander polynomial with the first non-vanishing higher Alexander polynomial.
(This talk is also a colloquium, 1:00pm Friday, Dec 4, 2015.)
Date received: September 29, 2015
Copyright © 2015 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 # cblw-05.