Small knots in some closed Haken 3-manifolds.
A closed orientable surface \(F\) embedded in a compact orientable 3-manifold is essential if \(F\) is a 2-sphere not bounding a 3-ball in \(M\). A manifold is called small if there is no closed essential surface embedded in it. In this paper, the author constructs infinitely many new examples of small 3-manifolds \(M\) with boundary a single torus. Under this construction the manifolds \(M\) also satisfy other properties, for instance, (i) there exists a small hyperbolic knot \(K\) in \(M\) (recall that a knot \(K\) in a closed 3-manifold \(M\) is called hyperbolic if \(M-N(K)\) admits a complete hyperbolic structure of finite volume) and (ii) the manifold \(M\) cannot be obtained by Dehn surgery on any knot in \(S^3\).
scientific article; zbMATH DE number 2037235
scientific article