Casson towers and filtrations of the smooth knot concordance group

Abstract

The n-solvable filtration $\{\mathcal{F}_n\}_{n=0}^\infty$ of the smooth knot concordance group (denoted by $\mathcal{C}$), due to Cochran-Orr-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 characterizations 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-Horn, the positive and negative filtrations, denoted by $\{\mathcal{P}_n\}_{n=0}^\infty$ and $\{\mathcal{N}_n\}_{n=0}^\infty$ respectively. In particular, we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball with only positive (resp. negative) kinks in the base-level kinky disk, then K is 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 the 4-ball, it bounds an embedded (symmetric) grope of height n+2, and is therefore, n-solvable (this also implies that topologically slice knots bound arbitrarily tall gropes in the 4-ball). We also define a variant of Casson towers and show that if K bounds a tower of type (2,n) in the 4-ball, it is n-solvable. If K bounds such a tower with only positive (resp. negative) kinks in the base-level kinky disk then K is in $\mathcal{P}_n$ (resp. $\mathcal{N}_n$). Our results show that either every knot which bounds a Casson tower of height three is topologically slice or there exists a knot which is not topologically slice but lies in each $\mathcal{F}_n$. We also give a 3-dimensional characterization, up to concordance, of knots which bound kinky disks in the 4-ball with only positive (resp. negative) kinks; such knots form a subset of $\mathcal{P}_0$ (resp. $\mathcal{N}_0$).