Knots on once-punctured torus fibers

We study knots that lie as essential simple closed curves on the fiber of a genus one fibered knot in S 3 . We determine certain surgery descriptions of these knots that enable estimates on volumes of these knots. We also develop an algorithm to list all closed essential surfaces in the complement of a given knot in this family. Relationships between the volumes of such knots and the surfaces in their exteriors is then examined.