A Phenomenological Approach to Interactive Knot Diagrams

TL;DR

This paper introduces a new phenomenological method for real-time, interactive manipulation of 2D knot diagrams directly through vector graphics, enhancing tools for knot theory visualization and analysis.

Contribution

It presents a novel approach to interactively manipulate knot diagrams without relying on 3D structures, enabling real-time interaction and invariant computation.

Findings

Developed a method for direct interaction with 2D knot diagrams.

Implemented a tool that allows real-time manipulation and invariant calculation.

Facilitated knot theory teaching and research with an accessible digital tool.

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.