A Phenomenological Approach to Interactive Knot Diagrams

TL;DR

The paper presents a 2D, topology-preserving, interactive knot-diagram editor that treats a knot as a planar path $\gamma(t)$ built from Bézier curves and maintains crossing topology frame-by-frame via an optimal transport mapping using a crossing-distance $d_\gamma$. The method enables real-time user manipulation with guarded Reidemeister-move validation and computes invariants such as the Alexander polynomial in situ, all implemented in an open-source web tool called Knottingham. It demonstrates the practicality of directly editing knot diagrams in vector graphics while preserving topology to the extent feasible for real-time interaction, and discusses both limitations and avenues for extending to links, performance, and invariants. The work advances interactive knot theory tooling with immediate visual feedback, educational applicability, and a foundation for computer-assisted knot equivalence exploration.

Abstract

Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly available tool to manipulate knot diagrams in a real-time, interactive way is yet to be developed. We introduce a method of operating on the geometry of the knot diagram itself without any underlying three-dimensional structure that can underpin such an application. This allows us to directly interact with vector graphics knot diagrams while at the same time computing knot invariants in ways proposed by previous work. An implementation of this method is provided.

Figures (5)

• Figure 1: A knot diagram produced by our implementation. Intersections are coloured according to their over-under-boolean.
• Figure 2: Inferring crossing booleans from a previous trefoil (top left) and a new curve (top right) manipulated by the user to a new trefoil (bottom). Knot diagrams were produced by our program.
• Figure 3: The Reidemeister Crimes. These are the illegal moves that we need to revert once the user attempts them. Above is a wrong version of Reidemeister 2, below a wrong version of Reidemeister 3 where the left side shows a tangle. Any combination of over-under-boolean on the bottom right would be illegal.
• Figure 4: The Reidemeister Moves. These preserve the topology, so we allow an operation if it can be identified as one of them.
• Figure 5: A screenshot of the implementation showing one representative of the Perko pair, pieced together by continuously differentiable Bézier curves with visible handles. Blue dots can be moved with the mouse. The Alexander polynomial correctly recognises the knot as entry $\texttt{1\_161}$ in the Rolfsen tablerolfsen2003knots. One application of the program could be to transform the diagram above into the representation given by the Knot Atlas knotatlas.