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4:00 pm Monday, August 26, 2013 Topology Seminar: Casson towers and filtrations of the smooth knot concordance groupby Arunima Ray (Rice) in HB 427- The n-solvable filtration {F_n} of the smooth knot concordance group, 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 its membership in various levels of the filtration. We show the counterpart of this result for two new filtrations of the knot concordance group due to Cochran-Harvey-Horn, the positive and negative filtrations denoted by {P_n} and {N_n} 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 disks, then K is in P_n (resp. 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. 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 P_n (resp. 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 F_n for all n which is not topologically slice.
Submitted by cochran@rice.edu |