A Geometric Approach to Simulating Knotting and Entanglement of Slender Objects

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Knotting and entanglement play a pivotal role in our daily lives (locs of hair, knitting, sailing, rockface climbing, surgical sutures, pocketed headphone cables, shoelaces). Knots are also pivotal in science (DNA, molecules) and engineering (polymers, textiles, nautical/aerospace). Despite their ubiquity and importance, our fundamental understanding of entanglement remains at its infancy, even for the most basic questions: (i) will this knot hold? (ii) what mechanism makes this knot so effective in one situation but not another? (iii) what factors are most likely to change its efficacy? (iv) under what conditions do certain knots spontaneously form (headphone cables in our pocket) or become undone (our shoelaces while running), and how can this be prevented? One avenue to understanding these important questions is to compute motion. Using a computer, we can predict how materials move or change shape due to their environment. When computers predict how materials move and deform, they help us to understand, predict, and more safely interact with the physical world. From billowing hair to swimming microorganisms to robotics, computing motion broadly impacts the arts, science and engineering. This research project will develop computational methods for computing the motions involved in knotting and entanglement. The project advances many aspects of this problem, from theoretical to applied. The research will address questions such as, what does it mean for a cable to become entangled, mathematically speaking? Can we design computer programs that are guaranteed to produce a trustworthy result for a knotting scenario, and do it fast? And if there are multitudes of fibres in a dense entangled network, how can we accelerate the computation while maintaining the quality of the resulting prediction? This project will help transfer these software tools to the Canadian visual special effects and games industries so that virtual characters can have realistic hair of all types, shapes, coarseness, and density. We will be better able to model the wide variety of hairstyles and types found in society. This research will also use the resulting computer software to study fundamental questions about entanglement. For instance, take a few thousand paperclips, bend them all into corkscrews, put them in a closed box, and shake: we expect a tangled mess. Try instead bending the paperclips into straight lines: we no longer expect them to cling to each other. What about for other shapes, such as the letters S, N, or G? Given a bent wire, can we predict its self-entanglement behaviour? This is a fundamental open question with significant ramifications for the design of new materials, understanding biology, and even engineering "programmable matter." The software produced by this project will be made freely available, enabling anyone to build on it to advance the arts, science and engineering.