Celebratio Mathematica — Gordon — Complete Bibliography

Complete Bibliography

Works connected to Danny M. C. Calegari

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D. Calegari and C. Gordon: “Knots with small rational genus,” Comment. Math. Helv. 88 : 1 (2013), pp. 85–130. MR 3008914 Zbl 1275.57019 ArXiv 0912.1843 article

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.