Tunnel number one knots, \(m\)-small knots and the Morimoto conjecture
The tunnel number \(t(K)\) of a knot \(K\) in the 3--sphere \(S^3\) is the least number of arcs that can be attached to \(K\) to produce a graph whose complement in \(S^3\) is a handlebody. Given two knots \(K_1\) and \(K_2\) in \(S^3\), the tunnel number of their connected sum satisfies \(t(K_1 \# K_2) \leq t(K_1) + t(K_2) + 1\). The original Morimoto conjecture asserted that equality held if and only if neither \(K_1\) nor \(K_2\) was \(\mu\)--primitive, meaning that their knot complements had a minimal primitive Heegard splitting. Counterexamples were found and the conjecture was revised to apply only to prime knots. (See [\textit{Y. Moriah}, Proceedings of the Technion workshop on Heegaard splittings, Haifa, Israel, summer 2005. Coventry: Geometry \& Topology Publications. Geometry and Topology Monographs 12, 191--232 (2007; Zbl 1138.57008)]) In the paper under review, the authors prove the conjecture when each of \(K_1\) and \(K_2\) either have tunnel number one or are \(m\)--small, meaning that their knot complement contains no meridional essential surface.
scientific article; zbMATH DE number 6218167