The shapes of physical trefoil knots

TL;DR

This study compares the equilibrium shapes of physical and ideal trefoil knots, revealing similarities and differences that could influence the strength and deformation of knotted elastic filaments.

Contribution

It provides the first detailed comparison between physical elastomeric trefoil knots and their ideal geometric models, highlighting the impact of elasticity on knot shape.

Findings

Physical knots show shape similarities to ideal models

Elastic deformation regions may indicate weak spots

Differences in curvature and contact regions identified

Abstract

We perform a compare-and-contrast investigation between the equilibrium shapes of physical and ideal trefoil knots, both in closed and open configurations. Ideal knots are purely geometric abstractions for the tightest configuration tied in a perfectly flexible, self-avoiding tube with an inextensible centerline and undeformable cross-sections. Here, we construct physical realizations of tight trefoil knots tied in an elastomeric rod, and use X-ray tomography and 3D finite element simulation for detailed characterization. Specifically, we evaluate the role of elasticity in dictating the physical knot's overall shape, self-contact regions, curvature profile, and cross-section deformation. We compare the shape of our elastic knots to prior computations of the corresponding ideal configurations. Our results on tight physical knots exhibit many similarities to their purely geometric counterparts, but also some striking dissimilarities that we examine in detail. These observations raise the hypothesis that regions of localized elastic deformation, not captured by the geometric models, could act as precursors for the weak spots that compromise the strength of knotted filaments.