A note on genus one fibered knots in lens spaces

Abstract

Supose that $Y$ is a lens space with $|H_1(Y; \mathbb{Z})|$ prime, and $Y$ does not contain a genus one fibered knot. We show that $Y$ contains a knot whose exterior is a once-punctured torus bundle if and only if $Y$ is the result of $p/q$-surgery on the trefoil. This partially answers a question posed by Ken Baker in a paper in which he gives a complete classification of genus one fibered knots contained in lens spaces. Combining Baker's classification with Moser's characterization of lens space surgeries on the trefoil, we generate an infinite family of lens spaces which do not contain any knot whose exterior is a once-punctured torus bundle.