Haptic Exploration of Mathematical Knots

We present a novel multi-modal haptic interface for sketching and exploring the structure and properties of mathematical knots. Our interface derives from the familiar pencil-and-paper process of drawing 2D knot diagrams to facilitate the creation and exploration of mathematical knots; however, with a touch-based interface, users can also leverage their physical intuition by seeing, touching, and feeling the knots. The pure haptic component provides an intuitive interaction model for exploring knots, focusing on resolving the apparent conflict between the continuous structure of the actual knot and the visual discontinuities at occlusion boundaries. The auditory component adds redundant cues that emphasize the traditional knot crossings, where the haptic proxy crosses a visual disruption in the graphics image. Our paradigm enhances and extends traditional 2D sketching methods by exploiting both touch and sound to assist in building clearer mental models of geometry such as knot structures.