Two-solvable and two-bipolar knots with large four-genera

Abstract

For every integer g, we construct a 2-solvable and 2-bipolar knot whose topological 4-genus is greater than g. Note that 2-solvable knots are in particular algebraically slice and have vanishing Casson-Gordon obstructions. Similarly all known smooth 4-genus bounds from gauge theory and Floer homology vanish for 2-bipolar knots. Moreover, our knots bound smoothly embedded height four gropes in $D^4$, an a priori stronger condition than being 2-solvable. We use new lower bounds for the 4-genus arising from $L^{(2)}$-signature defects associated to meta-metabelian representations of the fundamental group.