Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation
by
Christopher Herald
University of Nevada, Reno
Coauthors: Paul Kirk, Charles Livingston

One obstruction to topological sliceness developed by Kirk and Livingston (’99) is based on the twisted Alexander polynomial associated to a U(1) representation of the fundamental group of a cyclic cover of the knot complement. In this talk, we will discuss an identification of this twisted polynomial with a twisted polynomial associated to a metabelian representation of the knot group itself. This identification simplifies the calculation of the twisted polynomial significantly.

As applications of the more efficient method for calculating these, we prove that 16 (of the 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice, and we show one of the remaining two to be slice. We also show that the 24 mutants of the pretzel knot P(3, 7, 9, 11, 15) coming from permutations of the indices represent distinct concordance classes.

Paper reference: arXiv:0804.1355

Date received: May 1, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caxb-10.