Twist Deformation in Trefoil Knot

Knots are applied on various scales, from micro to macro-scales, such as polymers, DNA, shoelaces, and surgery, serving their unique mechanical properties. The shape of ideal knots has been extensively studied as knot theory in the field of mathematics. However, to predict the shape of physical knots, we need to disentangle the complex interplay between self-contact and friction of the rod, bending, and twist properties. Although the mechanics of physical knots have been investigated by many researchers in the last decade, the unified understanding of the deformation of physical knots is still 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. In the experiment, we apply twist at the rod (made of silicone elastomer) extremities to examine the role of twist in the mechanics of trefoil knots. As we twist the rod, the knot becomes either tightened or loosened, preserving the original three-fold symmetry, and then buckles and transits to the state with different symmetry at critical angles. The transition would be triggered by the mechanical instability, where the imposed twist energy is converted into the bending energy. In the presentation, we will discuss the results in more detail, together with the simulation results.