## 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.

### MSC:

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

Zbl 0747.57001
Full Text: