Counting Genus One Fibered Knots in Lens Spaces

The braid axis of a closed 3-braid lifts to a genus one fibered knot in the double cover of S-3 branched over the closed braid. Every genus one fibered knot in a 3-manifold may be obtained in this way. Using this perspective, we answer a question of Morimoto about the number of genus one fibered knots in lens spaces. We determine the number of genus one fibered knots up to homeomorphism and up to isotopy in any given lens space. This number is 3 in the case of the lens space L(4, 1), 2 for the lens spaces L (m, 1) with m > 0 and m not equal 4, and at most 1 otherwise. Furthermore, each homeomorphism equivalence class in a lens space is realized by at most two isotopy classes.