Cabling in terms of immersed curves

TL;DR

This paper interprets Heegaard Floer homology for manifolds with torus boundary using immersed curves, and provides a formula describing how these curves change under cabling operations, connecting knot invariants with geometric representations.

Contribution

It introduces a new geometric interpretation of knot Floer homology via immersed curves and derives a formula for their transformation under cabling, advancing the understanding of Floer invariants.

Findings

Immersed curves encode knot Floer homology.

A formula for the behavior of curves under cabling is established.

The approach links bordered Floer homology with geometric models.

Abstract

In joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves in a punctured torus. In particular, knot Floer homology is captured by this invariant. Appealing to earlier work of the authors on bordered Floer homology, we give a formula for the behaviour of these immersed curves under cabling.