A classification of pairs of disjoint nonparallel primitives in the boundary of a genus two handlebody

TL;DR

This paper classifies pairs of disjoint nonparallel primitive curves on the boundary of a genus two handlebody, revealing they either lie on opposite ends of specific product structures or bound certain essential annuli, depending on their properties.

Contribution

It provides a comprehensive classification of such pairs, distinguishing cases based on their position and whether one is a proper power of a primitive.

Findings

Pairs lie on opposite ends of a product or twisted product structure.

If one curve is a proper power, they are separated by a disk or bound an essential annulus.

Classification simplifies understanding of primitive curve configurations in genus two handlebodies.

Abstract

Embeddings of pairs of disjoint nonparallel primitive simple closed curves in the boundary of a genus two handlebody are classified. Briefly, two disjoint primitives either lie on opposite ends of a product $F \boldsymbol{\times} I$, or they lie on opposite ends of a kind of "twisted" product $F \widetilde{\boldsymbol{\times}} I$, where $F$ is a once-punctured torus. If one of the curves is a proper power of a primitive, the situation is simpler. Either the curves lie on opposite sides of a separating disk in the handlebody, or they bound a nonseparating essential annulus in the handlebody.