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

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

2009-10-16

arxiv.org/abs/0910.3038v1

Mathematics

Geometric Topology

13 pages, 9 figures

Scientific paper

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.

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