JournalscmhVol. 88, No. 1pp. 85–130

Knots with small rational genus

  • Danny Calegari

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

    University of Texas at Austin, USA
Knots with small rational genus cover
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If KK is a rationally null-homologous knot in a 33-manifold MM, the rational genus of KK is the infimum of χ(S)/2p-\chi(S)/2p over all embedded orientable surfaces SS in the complement of KK whose boundary wraps pp times around KK for some pp (hereafter: SS is a pp-Seifert surface for KK). 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 MM – and give a complete classification. Our arguments are a mixture of hyperbolic geometry, combinatorics, and a careful study of the interaction of small pp-Seifert surfaces with essential subsurfaces in MM 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