In this paper we present a mathematical knot/braid diagram interface that exploits 3D computer graphics, interactive visualization, and multi-touch technology to enhance one's intuitive experience with mathematical theory of knots. Our interaction model is based on the clever but simple geometric construction named the Reide-meister moves, which allows 3D topological manipulations using rather simple 2D moves. Multi-touch interfaces can provide a natural way for us to interact with the extra degrees of freedom that characterize knots' mathematical, physical, and arithmetic properties. Relative to a specialized mouse-driven interface, the proposed multi- touch interface is easier and more intuitive to learn, and our pilot study shows that knot and braid manipulation with multi- touch is much faster and more efficient. All these combine to show that interactive computer graphics methods and computer interfaces can be used to construct virtual manipulatives and meet the challenge of exploring abstract mathematical worlds.