Knots with small rational genus

Abstract

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.