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

## TL;DR

This paper constructs specific 2-solvable and 2-bipolar knots with arbitrarily large topological 4-genus, using new $L^{(2)}$-signature techniques to establish lower bounds, advancing understanding of knot concordance.

## Contribution

It introduces a method to produce knots with large 4-genus within the classes of 2-solvable and 2-bipolar knots, using novel $L^{(2)}$-signature invariants.

## Key findings

- Existence of knots with arbitrarily large 4-genus in these classes.
- New lower bounds for 4-genus from $L^{(2)}$-signatures.
- Knots bound smooth gropes of height four in $D^4$.

## 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.