Dehn surgery on the figure 8 knot: Immersed surfaces. (English) Zbl 0926.57006

Let \(M_8\) denote the complement of the figure eight knot in \(S^3\), and \( M_{(p,q)}\) the closed orientable 3-manifold obtained from \(M_8\) by \((p,q)\)-Dehn surgery. Thurston [The geometry and topology of 3-manifolds, Princeton University Lecture Notes, 1978] showed that all but finitely surgeries on the figure eight knot yield non-Haken hyperbolic 3-manifolds. Therefore, no closed, incompressible surfaces exist in these manifolds. It is conjectured that every closed hyperbolic 3-manifold contains an immersed incompressible surface of negative Euler characteristic. It is known that about 70% of surgeries on the figure eight knot give 3-manifolds which contain immersed incompressible surfaces of genus greater than 1. One of the main results in the article under review improves this to about 80% by showing that all even surgeries give manifolds containing such a surface. It is shown in [M. D. Baker, Pac. J. Math. 150, No. 2, 215-228 (1991; Zbl 0747.57001)] that for every \(k\), \( M_{(3k,q)}\) has virtually \({\mathbb Z}\)-representable fundamental group, unless \( q=k\pm 1\). Another main result of the paper under review partially deals with half of the exceptional cases in Baker’s result by showing that for every \(k,q\neq 0\), \(M_{(6k,q)}\) is virtually Haken.


57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
57R65 Surgery and handlebodies


Zbl 0747.57001
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