We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which do not 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.
"On Knots with trivial Alexander polynomial." J. Differential Geom. 67 (1) 167 - 193, May 2004. https://doi.org/10.4310/jdg/1099587731