A Suggestive Interface for Untangling Mathematical Knots

In this paper we present a user-friendly sketching-based suggestive interface for untangling mathematical knots with complicated structures. Rather than treating mathematical knots as if they were 3D ropes, our interface is designed to assist the user to interact with knots with the right sequence of mathematically legal moves. Our knot interface allows one to sketch and untangle knots by proposing the Reidemeister moves, and can guide the user to untangle mathematical knots to the fewest possible number of crossings by suggesting the moves needed. The system highlights parts of the knot where the Reidemeister moves are applicable, suggests the possible moves, and constrains the user's drawing to legal moves only. This ongoing suggestion is based on a Reidemeister move analyzer, that reads the evolving knot in its Gauss code and predicts the needed Reidemeister moves towards the fewest possible number of crossings. For our principal test case of mathematical knot diagrams, this for the first time permits us to visualize, analyze, and deform them in a mathematical visual interface. In addition, understanding of a fairly long mathematical deformation sequence in our interface can be aided by visual analysis and comparison over the identified "key moments" where only critical changes occur in the sequence. Our knot interface allows users to track and trace mathematical knot deformation with a significantly reduced number of visual frames containing only the Reidemeister moves being applied. All these combine to allow a much cleaner exploratory interface for us to analyze and study mathematical knots and their dynamics in topological space.