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

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 $F\times I$, or they lie on opposite ends of a kind of "twisted" product $F\tilde{\times }I$, 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.

13 pages, 9 figures

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