Topologically slice knots of smooth concordance order two

TL;DR

This paper demonstrates the existence of an infinite subgroup within the smooth concordance group generated by topologically slice knots of order two, using Heegaard-Floer theory, and shows these knots cannot have Alexander polynomial one.

Contribution

It introduces a new infinite subgroup of the smooth concordance group generated by specific topologically slice knots of order two, expanding understanding of knot concordance.

Findings

Existence of an infinite subgroup generated by topologically slice knots of order two

No nontrivial element in this subgroup has Alexander polynomial one

Application of Heegaard-Floer theory to establish these results

Abstract

The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to 4-manifolds. Here, as an application of Ozsvath and Szabo's Heegaard-Floer theory, we show the existence of an infinite subgroup of the smooth concordance group generated by topologically slice knots of concordance order two. In addition, no nontrivial element in this subgroup can be represented by a knot with Alexander polynomial one.