A note on cobordisms of algebraic knots

TL;DR

This paper explores how Heegaard Floer homology and the concordance invariant ν+ can be used to analyze smooth cobordisms of algebraic knots and complex deformations of cusp singularities, providing explicit formulas and subadditivity results.

Contribution

It introduces explicit formulas for ν+ in the case of L-space knots and establishes subadditivity of ν+ for general algebraic knots, advancing understanding of knot cobordisms.

Findings

Explicit formula for ν+ for L-space knots

Subadditivity of ν+ for algebraic knots

Insights into complex deformations of cusp singularities

Abstract

In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant $\nu^+$: we study its behaviour with respect to connected sums, providing an explicit formula in the case of L-space knots and proving subadditivity in general.