Tunnel number, connected sum and meridional essential surfaces
Let \(K_1,\ldots,K_n\) be knots in \(S^3\) and \(t(K_i)\), \(i=1,\ldots,n\) the tunnel numbers of them. \textit{K. Morimoto} and \textit{J. Schultens} [Proc. Am. Math. Soc. 128, No. 1, 269-278 (2000; Zbl 0928.57003)] showed that if none of the knot exteriors \(E(K_1),\ldots,E(K_n)\) contains closed essential surfaces, then \(t(K_1\#\ldots\#K_n)\geq t(K_1)+\ldots+t(K_n)\). This result is here generalized to show that, for given knots \(K_1,\ldots,K_n\) in orientable closed \(3\)-manifolds \(M_1,\ldots,M_n\) respectively, if none of those \(3\)-manifolds has lens space summands and none of those knot exteriors contains meridional essential surfaces, then the inequality above holds. The proof depends on the concept of strongly irreducible Heegaard splittings which has been developed by \textit{M. Scharlemann} and \textit{A. Thompson} [Contemp. Math. 164, 231-238 (1994; Zbl 0818.57013)] together with a concept of an amalgamation of Heegaard splittings due to \textit{J. Schultens} [Proc. Lond. Math. Soc., III. Ser. 67, No. 2, 425-448 (1993; Zbl 0789.57012)].
strongly irreducible Heegaard splittings
scientific article; zbMATH DE number 1436046