The Easy Knot: Computationally Implementing Reidemeister Moves for Unknotting

Picture a very large bundle of knotted string. Could you untie it into a straight lace? How many twists and loops might it take to do so? Although simple in concept, the problem of determining whether a knot can be untangled within a bounded number of moves, and whether it represents another knot, has been proven to be NP-hard. Existing tools for exploring knots primarily focus on identifying knots through invariants and visualizations, making transformations largely manual and difficult to execute. This paper explores the components necessary to reframe the problem through a novel data structure called the Easy Knot. This structure is designed to parametrize knots in a way that enables automated generation, untangling, and comparison, thereby providing a new perspective on the computational complexity of knot simplification and equivalence.