Unknotting number and cabling

TL;DR

This paper provides a new lower bound on the unknotting number of cable knots based on their winding number, utilizing advanced knot Floer homology techniques.

Contribution

It introduces a novel lower bound for the unknotting number of cable and iterated cable knots using knot Floer homology and immersed curve computations.

Findings

Lower bound on unknotting number in terms of winding number

Application of knot Floer homology to cable knots

Extension to iterated cable knots

Abstract

The unknotting number of knots is a difficult quantity to compute, and even its behavior under basic satelliting operations is not understood. We establish a lower bound on the unknotting number of cable knots and iterated cable knots purely in terms of the winding number of the pattern. The proof uses Alishahi-Eftekhary's bounds on unknotting number from knot Floer homology together with Hanselman-Watson's computation of the knot Floer homology of cables in terms of immersed curves in the punctured torus.