On the super additivity of tunnel number of knots
Let \(K\) be a knot in \(S^{3}\), \(N(K)\) the regular neighborhood of \(K\) in \(S^{3}\), \(E(K) = \text{ cl}(S^{3}-N(K))\) the exterior of \(K\). Then the tunnel number of \(K\), \(t(K)\), is the minimal number of mutual disjoint arcs \(\gamma_{1}, \gamma_{2},\dots,\gamma_{t}\) properly embedded in \(E(K)\) such that \[ \text{ cl}(E(K) -N(\partial E(K)\cup \gamma_{1} \cup \gamma_{2}\cup \dots \cup \gamma_{t})) \] is a handlebody, where \(N(\partial E(K)\cup \gamma_{1} \cup \gamma_{2}\cup \dots \cup \gamma_{t})\) is a regular neighborhood of \(\partial E(K)\cup \gamma_{1} \cup \gamma_{2}\cup \dots \cup \gamma_{t}\) in \(E(K)\). Let \(b\) be a positive integer. Then, for a given knot \(K\) in \(S^{3}\), it is well known that there is a Heegaard splitting \((V_{1},V_{2})\) of \(S^{3}\) such that K intersects \(V_{i}\) in a \(b\)-string trivial arc system in \(V_{i}\) for both \(i=1,2\). We call such a decomposition of \((S^{3},K)\) a \(b\)-bridge decomposition with genus \(g\). We define the \(b\)-bridge genus of \(K\), denoted by \(g_{b}(K)\), as the minimal genus among all \(b\)-bridge decompositions of \((S^{3},K)\). In this paper we consider the special case when \(b=1\). In this case the \(1\)-bridge genus and the tunnel number are related by the following inequality: \(g_{1}(K)-1 \leq t(K) \leq g_{1}(K)\). We say a surface \(F\) properly embedded in a compact 3-manifold is essential if \(F\) is incompressible and not \(\partial\)-parallel. Let \(F\) be a surface properly embedded in the knot exterior \(E(K)\). Then \(F\) is meridional if \(\partial F \neq \emptyset\) and each component of \(\partial F\) is a meridian of \(K\). We say a knot \(K\) in \(S^{3}\) is meridionally small if \(E(K)\) contains no meridional essential surfaces. The main result in the paper is the following: We denote the connected sum of two knots \(K_{1}\), \(K_{2}\) by \(K_{1} \sharp K_{2}\). Let \(K_{1}\), \(K_{2}\) be knots in \(S^{3}\), and suppose that both \(K_{1}\) and \(K_{2}\) are meridionally small. Then \(t(K_{1} \sharp K_{2}) = t(K_{1}) + t(K_{2})+1\) if and only if \(g_{1}(K_{i})= t(K_{i})+1\) for both \(i=1,2\). This result can be generalized from knots in \(S^{3}\) to knots in closed orientable 3-manifolds that do not contain lens space summands.
scientific article; zbMATH DE number 1502096