Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation

Details Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation

2008-04-08

arxiv.org/abs/0804.1355v1

Mathematics

Geometric Topology

24 pages, 1 figure

Scientific paper

Given a knot complement X and its p-fold cyclic cover X_p, we identify twisted polynomials associated to 1-dimensional linear representations of the fundamental group of X_p with twisted polynomials associated to related p-dimensional linear representations of the fundamental group of X. This provides a simpler and faster algorithm to compute these twisted polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P(3,7,9,11,15), corresponding to permutations of (7,9,11,15), represent distinct concordance classes.

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