While rope is arguably a simpler system to simulate than cloth, the real-time simulation of rope, and knot tying in particular, raise unique and difficult issues in contact detection and management. Some practical knots can only be achieved by complicated crossings of the rope, yielding multiple simultaneous contacts, especially when the rope is pulled tight. This paper describes a simulator allowing a user to grasp and smoothly manipulate a virtual rope and to tie arbitrary knots, including knots around other objects, in real-time. One component of the simulator precisely detects selfcollisions in the rope, as well as collisions with other objects. Another component manages collisions to prevent penetration, while making the rope slide with some friction along itself and other objects, so that knots can be pulled tight in believable manner. An additional module uses recent results from knot theory to identify which topological knots have been tied, also in real-time. This work was motivated by surgical suturing, but simulation in other domains, such as sailing and rock climbing, could benefit from it.