Stable elastic knots with no self-contact

Knots are widespread, universal physical structures, from shoelaces to Celtic decoration to the many variants familiar to sailors. They are often simple to construct and aesthetically appealing, yet remain topologically and mechanically quite complex. Knots are also common in biopolymers such as DNA and proteins, with numerous and significant biological implications.

While self-contact is an inevitable feature of tight knots, here we go the other direction and ask whether a knotted filament with zero points of self-contact may be realized physically. Our focus is on the simple hand-held experiment of an elastic rod bent into a trefoil knot, with the ends held clamped. The question we consider is whether there exist stable configurations for which there are no points of self-contact. This idea can be fairly easily replicated with a thin strip of paper, but is more difficult or even impossible with a flexible wire. We search for such configurations within the space of three tuning parameters related to the degrees of freedom in the simple experiment. Mathematically, we show, both within standard Kirchhoff theory as well within an elastic strip theory, that stable and contact-free knotted configurations can be found, and we classify the corresponding parametric regions. Numerical results are complemented with an asymptotic analysis that demonstrates the presence of knots near the doubly-covered ring. In the case of the strip model, quantitative experiments of the region of good knots are also provided to validate the theory.
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