Casson towers and filtrations of the smooth knot concordance group

Abstract

The $n$–solvable filtration ${\left\{{\mathcal{ℱ}}_n\right\}}_{n=0}^{\infty }$ of the smooth knot concordance group (denoted by $\mathcal{C}$) due to Cochran, Orr and Teichner has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric attributes of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of $\mathcal{C}$ due to Cochran, Harvey and Horn; the positive and negative filtrations, denoted by ${\left\{{\mathcal{P}}_n\right\}}_{n=0}^{\infty }$ and ${\left\{{\mathcal{N}}_n\right\}}_{n=0}^{\infty }$ respectively. In particular, we show that if a knot $K$ bounds a Casson tower of height $n+2$ in $B^4$ with only positive (resp. negative) kinks in the base-level kinky disk, then $K\in {\mathcal{P}}_n$ (resp. ${\mathcal{N}}_n$). En route to this result we show that if a knot $K$ bounds a Casson tower of height $n+2$ in $B^4$, it bounds an embedded (symmetric) grope of height $n+2$ and is therefore $n$–solvable. We also define a variant of Casson towers and show that if $K$ bounds a tower of type $\left(2,n\right)$ in $B^4$, it is $n$–solvable. If $K$ bounds such a tower with only positive (resp. negative) kinks in the base-level kinky disk then $K\in {\mathcal{P}}_n$ (resp. $K\in {\mathcal{N}}_n$). Our results show that either every knot which bounds a Casson tower of height three is topologically slice or there exists a knot in $\bigcap {\mathcal{ℱ}}_n$ which is not topologically slice. We also give a $3$–dimensional characterization, up to concordance, of knots which bound kinky disks in $B^4$ with only positive (resp. negative) kinks; such knots form a subset of ${\mathcal{P}}_0$ (resp. ${\mathcal{N}}_0$).