On Knots with trivial Alexander polynomial

Details On Knots with trivial Alexander polynomial On Knots with trivial Alexander polynomial

2002-06-04

arxiv.org/abs/math/0206023v3

Mathematics

Geometric Topology

LaTeX, 15 pages with 23 figures

Scientific paper

We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which don't have a Seifert surface whose genus equals the rank of the Seifert form. This is one of the first applications of the Kontsevich integral to intrinsically 3-dimensional questions in topology. Our examples contradict a lemma of Mike Freedman, and we explain what went wrong in his argument and why the mistake is irrelevant for topological knot concordance. References updated.

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