CASSON TOWERS AND FILTRATIONS OF THE SMOOTH KNOT CONCORDANCE GROUP

Abstract. The n–solvable filtration {Fn}∞n=0 of the smooth knot concor-dance group (denoted by C), due to Cochran–Orr–Teichner, has been instru-mental 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 C due to Cochran–Harvey–Horn, the positive and negative filtrations, denoted by {Pn}∞n=0 and {Nn}∞n=0 respectively. In particular, we show that if a knot K bounds a Casson tower of height n+ 2 in B4 with only positive (resp. negative) kinks in the base-level kinky disk, then K ∈ Pn (resp. Nn). En route to this result we show that if a knot K bounds a Casson tower of height n + 2 in B4, 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 (2, n) in B4, it is n–solvable. If K bounds such a tower with only positive (resp. negative) kinks in the base-level kinky disk then K ∈ Pn (resp. K ∈ Nn). Our results show that either every knot which bounds a Casson tower of height three is topologically slice or there exists a knot in ⋂Fn which is not topologically slice. We also give a 3–dimensional characterization, up to concordance, of knots which bound kinky disks in B4 with only positive (resp. negative) kinks; such knots form a subset of P0 (resp. N0). 1