Filtering smooth concordance classes of topologically slice knots

Abstract

We propose and analyze a structure with which to organize the difference between a knot in S 3 bounding a topologically embedded 2-disk in B 4 and it bounding a smoothly embedded disk. The n-solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n-solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, fB n g, that is simultaneously a refinement of the n-solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each B n =B nC1 has infinite rank. But our primary interest is in the induced filtration, fT n g, on the subgroup, T , of knots that are topologically slice. We prove that T =T 0 is large, detected by gauge-theoretic invariants and the , s , -invariants, while the nontriviality of T 0 =T 1 can be detected by certain d -invariants. All of these concordance obstructions vanish for knots in T 1 . Nonetheless, going beyond this, our main result is that T 1 =T 2 has positive rank. Moreover under a "weak homotopy-ribbon" condition, we show that each T n =T nC1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity. 57M25

Covering link calculus and the bipolar filtration of topologically slice links

“…We do not know whether the Rasmussen s -invariant obstruction in [9] has a Z .p/ -homology analogue. Even the following weaker question is still left open.…”

“…negative, bipolar) introduced in [9,Sections 4,5,6] generalize to their Z .p/ -homology analogues. For those who are not familiar with these results, we spell out the statements.…”

“…negative, bipolar) cases given in [9]. Remark 2.9 Note that Theorem 2.8 also follows from our Covering positon theorem 3.2 that is stated and proved in Section 3: if we take a p a -fold branched cover M of S 3 along K , then by Theorem 3.2, M bounds a Z .p/ -homology 0-position, to which a result of Ozsváth-Szabó [21] applies to allow us to conclude that the dinvariant of M with appropriate spin c -structures is non-positive.…”

“…In order to understand the structure of topologically slice knots and links, T Cochran, S Harvey and P Horn introduced a framework for the study of smooth concordance in their recent paper [9]. This is an intriguing attempt at describing a global picture of the world of topologically slice links.…”

“…In particular, for 1-bipolar knots the following obstructions to being slice vanish: the -invariant and the -invariant from Heegaard Floer Knot homology, the s -invariant from the Khovanov homology (and consequently the Thurston-Bennequin invariant), and the Heegaard Floer correction term d -invariant of˙1-surgery manifolds and prime power fold cyclic branched covers, as well as gauge-theoretic obstructions derived from work of Fintushel-Stern, Furuta, Endo and Kirk-Hedden. For more details the reader is referred to [9].…”