Topologically slice knots with nontrivial Alexander polynomial

Abstract

Let C T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C ∆ be the subgroup generated by knots with trivial Alexander polynomial. We prove C T /C ∆ is infinitely generated. Our methods reveal a similar structure in the 3-dimensional rational spin bordism group, and lead to the construction of links that are topologically, but not smoothly, concordant to boundary links.

Concordance to links with unknotted components

“…It was observed in Hedden-Livingston-Ruberman [17] that the Ozsváth-Szabó dinvariant [21] gives an obstruction to being smoothly Q-homology cobordant to an integral homology sphere. To simplify the discussion of Spin c structures, we restrict to the case of Z 2 -homology spheres and homology cobordisms.…”

“…In [17], a calculation of d -invariants is used to elucidate the structure of smooth rational homology cobordism group modulo integral homology spheres. In this section, we investigate the more general case of Z p -homology spheres instead of integral homology spheres.…”

“…In particular the main ingredients are the following: covering link calculus as used in Cochran-Orr [12] and formulated in Cha-Kim [4] (see also ChaLivingston [8], Van Cott [22] and Levine [19]), invariants of rational knot concordance (see Cha [3] and also Cochran-Orr [12] and Cha-Ko [6]), and the d -invariant (or correction term) of Heegaard Floer homology (see Ozsváth-Szabó [21], and also JabukaNaik [18], Grigsby-Ruberman-Strle [16] and Hedden-Livingston-Ruberman [17]). …”

Covering link calculus and the bipolar filtration of topologically slice links

“…Consequently, for links with any given number of components, the filtration is non-trivial at level n for each n Ä 1. The knots of HeddenLivingston-Ruberman from [16], which were the first examples of topologically slice knots which are not smoothly concordant to knots with Alexander polynomial one, are also nontrivial in T =T 0 .…”

A family of non-split topologically slice links with arbitrarily large smooth slice genus

“…The study of smoothly and topologically slice links is closely connected with the study of smooth and topological 4-manifolds; e.g. any knot which is topologically slice but not smoothly slice [End95,Gom86,HK12,HLR12,Hom14]) gives rise to an exotic copy of R 4 [GS99, Exercise 9.4.23].…”