Twist deformation of physical trefoil knots

Abstract

Knots across various length scales, from micro to macro-scales, such as polymers, DNA, shoelaces, and surgery, serving their unique mechanical properties. The shapes of ideal knots have been extensively studied in the context of knot theory, while those of physical knots have only been recently discussed in the literature. The complex interplay of elasticity and geometry, such as bending, twisting, and contact, needs to be disentangled to predict their deformation. Still, the unified understanding of the deformation of physical knots is insufficient. Here, we focus on the trefoil knot, a closed knot with a nontrivial topology, and study the relationship between the shapes of the trefoil knot and applied physical twists, combining experiments and simulations. As we twist the elastomeric rod, the knot becomes either tightened or loosened, preserving the original three-fold symmetry, and then buckles and exhibits symmetry breaking at critical angles. The curvature profiles computed through the X-ray tomography ($\mu$CT) analysis also exhibit similar symmetry breaking. The transition would be triggered by the mechanical instability, where the imposed twist energy is converted into the bending energy. The phase transition observed here is analogous to the classical buckling phenomena of elastic rings known as the Michell instability. We find that the twist buckling instability of the trefoil knot results from the interplay of bending, twisting, and contact properties of the rod. In other words, the buckling of the knot is predictable based on the elasticity and geometry of rods, which would be useful in avoiding or even utilizing their buckling in practical engineering applications such as surgery and the shipping industry.