We compute the almost-sure Hausdorff dimension of the double points of chordal SLE$_\kappa$ for $\kappa > 4$, confirming a prediction of Duplantier--Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of chordal SLE$_\kappa$ for $\kappa > 4$ as well as analogous dimensions for the radial and whole-plane SLE$_\kappa(\rho)$ processes for $\kappa > 0$. We derive these facts as consequences of a more general result in which we compute the dimension of the intersection of two flow lines of the formal vector field $e^{ih/\chi}$, where $h$ is a Gaussian free field and $\chi > 0$, of different angles with each other and with the domain boundary.