Knot concordances and alternating knots

TL;DR

This paper constructs an infinitely generated free subgroup within the smooth knot concordance group, where no nontrivial element is represented by an alternating knot, yet all are topologically slice.

Contribution

It introduces a new subgroup in the knot concordance group with unique properties related to alternating and topologically slice knots.

Findings

Existence of an infinitely generated free subgroup with specific properties.

No nontrivial element in this subgroup is represented by an alternating knot.

Every element in the subgroup is topologically slice.

Abstract

There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every element is represented by a topologically slice knot.