On the additivity of \(h\)-genus of knots (Q1333598)

The \(h\)-genus of a knot \(K\) in \(S^3\) is the minimum genus \(h(K)\) of all Heegaard surfaces in \(S^3\) containing \(K\). Thus \(h(K)=1\) if and only if \(K\) is a (non-trivial) torus knot. The author shows that if the \(h\)-genus of a connected sum of two non-trivial knots \(K_1\), \(K_2\) is 2, then \(K_1\) and \(K_2\) are torus knots. On the other hand he gives examples to show that in general the \(h\)-genus is not additive with respect to connected sums.