Torsion in the knot concordance group and cabling

TL;DR

This paper introduces a new mod 2 invariant in the knot concordance group using involutive knot Floer homology, demonstrating that certain cables of knots have infinite order and constructing knots that are rationally slice but not slice.

Contribution

It defines a novel mod 2 invariant on the torsion subgroup of the knot concordance group and analyzes the properties of cable knots related to this invariant.

Findings

Certain cable knots have infinite order in the concordance group.

Infinitely many cable knots are linearly independent in the concordance group.

Constructs an infinite family of knots that are strongly rationally slice but not slice.

Abstract

We define a nontrivial mod 2 valued additive concordance invariant defined on the torsion subgroup of the knot concordance group using involutive knot Floer package. For knots not contained in its kernel, we prove that their iterated $(\text{odd},1)$-cables have infinite order in the concordance group and, among them, infinitely many are linearly independent. Furthermore, by taking $(2,1)$-cables of the aforementioned knots, we present an infinite family of knots which are strongly rationally slice but not slice.