Mathematical Research Letters

Volume 28 (2021)

Number 2

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

Pages: 331 – 382



Jae Choon Cha (Department of Mathematics, POSTECH, Pohang Gyeongbuk, South Korea; and School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea)

Allison N. Miller (Department of Mathematics, Rice University, Houston, Texas, U.S.A.)

Mark Powell (Department of Mathematical Sciences, Durham University, Durham, United Kingdom)


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.

The first author was partly supported by NRF grant 2019R1A3B206 7839.

Received 16 April 2019

Accepted 20 March 2020

Published 13 May 2021