Abstract: The real-time simulation of rope, and knot tying in particular, raises 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 graphical simulator that allows a user to grasp and smoothly manipulate a virtual rope and to tie arbitrary knots, including knots around other objects, in real time. A first component of the simulator computes the global configuration of the rope based on user interactions. Another component of the simulator precisely detects self-collisions in the rope as well as collisions with other objects. Finally, a third 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 a realistic manner. An additional module uses recent results from knot theory to identify, also in real time, which topological knots have been tied. This work was motivated by surgical suturing, but simulation in other domains, such as sailing and rock climbing, could also benefit from it.