# Knots with small rational genus

### Danny Calegari

California Institute of Technology, Pasadena, United States### Cameron Gordon

University of Texas at Austin, USA

## 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.

## Cite this article

Danny Calegari, Cameron Gordon, Knots with small rational genus. Comment. Math. Helv. 88 (2013), no. 1, pp. 85–130

DOI 10.4171/CMH/279