Knots with small rational genus

If K is a rationally null-homologous knot in a 3-manifold M, the rational genus of K is the infimum of -\chi(S)/2p over all embedded orientable surfaces S in the complement of K whose boundary wraps p times around K for some p (hereafter: S is a p-Seifert surface for K). Knots with very small rational genus can be constructed by "generic" Dehn filling, and are therefore extremely plentiful. In this paper we show that knots with rational genus less than 1/402 are all geometric -- i.e. they may be isotoped into a special form with respect to the geometric decomposition of M -- and give a complete classification. Our arguments are a mixture of hyperbolic geometry, combinatorics, and a careful study of the interaction of small p-Seifert surfaces with essential subsurfaces in M of non-negative Euler characteristic.