Commentarii Mathematici Helvetici


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Volume 88, Issue 1, 2013, pp. 85–130
DOI: 10.4171/CMH/279

Published online: 2013-01-07

Knots with small rational genus

Danny Calegari[1] and Cameron Gordon[2]

(1) California Institute of Technology, Pasadena, United States
(2) University of Texas at Austin, USA

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.

Keywords: Knots, rational genus, stable commutator length, Thurston norm, Berge conjecture

Calegari Danny, Gordon Cameron: Knots with small rational genus. Comment. Math. Helv. 88 (2013), 85-130. doi: 10.4171/CMH/279