Analytic knots, satellites and the 4-ball genus

TL;DR

This paper establishes a sharp lower bound for the 4-ball genus of analytic links in a tubular neighborhood of smoothly analytic knots, connecting complex geometry with knot theory.

Contribution

It introduces a new lower bound for the 4-ball genus of analytic links near smoothly analytic knots, advancing understanding of complex curves bounded by knots.

Findings

Provides a sharp lower bound for the 4-ball genus

Connects complex geometry with knot theory

Enhances understanding of analytic links in complex balls

Abstract

Call a smooth knot (or smooth link) in the unit sphere in $\mathbb{C}^2$ analytic (respectively, smoothly analytic) if it bounds a complex curve (respectively, a smooth complex curve) in the complex ball. Let $K$ be a smoothly analytic knot. For a small tubular neighbourhood of $K$ we give a sharp lower bound for the 4-ball genus of analytic links $L$ contained in it.