Concordance genus and the solvable filtration

The concordance genus is given by minimizing the 3-genus amongst all knots in a concordance class.  In the 80's Casson and Nakanishi used the Alexander polynomial to find knots for which the concordance genus does not agree with the 4-genus.  In 2004 Livingston used Casson-Gordon invariants to generate similar examples which cannot be seen by the Alexander polynomial.  This talk will continue that trend, finding an increasingly subtle sequence of infinite families of knots for which the concordance genus differs from the 4-genus.  As an added feature the knots presented today will have the property that no non-trivial linear combination can be written as a linear combination of knots with low concordance genus.