The Plain Sphere Number of a Link

Let $L$ be a link in $S^3$. We consider the class of meridional presentations for ${\pi }_1(S^3\mathrm{\backslash }L)$ in which the relations are witnessed by embedded two-spheres which can be represented simultaneously in a fixed diagram of $L$, analogously to decomposition spheres studied by Cromwell, Menasco and others. Wirtinger relations are witnessed by such spheres and the Wirtinger presentation is a special case of the ones we study. We prove that the smallest number of generators of ${\pi }_1(S^3\mathrm{\backslash }L)$ over all such presentations equals the bridge number of $L$.