A note on cables and the involutive concordance invariants

TL;DR

This paper derives a formula for involutive concordance invariants of cabled knots, showing certain cables are not smoothly slice and providing new bounds for their unknotting number, advancing knot theory understanding.

Contribution

It introduces a new formula linking cabled knots' invariants to those of their components, with implications for sliceness and unknotting bounds.

Findings

Iterated cables with parameters (odd,1) are not smoothly slice if invariants are nonzero.

New bounds for unknotting number of cabled knots are established.

The formula improves understanding of knot invariants and their applications.

Abstract

We prove a formula for the involutive concordance invariants of the cabled knots in terms of that of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer homology.