Snap buckling in overhand knots

We report a snap buckling process in overhand knots: when an overhand knot tied in elastic rods is tightened, it can undergo a sudden change in shape. We study this buckling process through a combination of discrete differential geometry (DDG)-based simulations and tabletop experiments. The onset of snap buckling is explored as a function of the topology of the knot, the rod geometry, and friction. In our setup, the two open ends of the overhand knot are slowly pulled which eventually leads to snap buckling in the closed-loop of the knot. We call this phenomenon “inversion” since the loop appears to dramatically move from its current position to the other side of the knot. A numerical framework is implemented with a combination of discrete elastic rods and a constraint-based method for frictional contact to explore the inversion in overhand knots. The numerical simulation can robustly capture the inversion in the knot and is found to be in good agreement with experimental results. In order to gain physical insight into the process, we employ scaling analysis on elastic energies and investigate the role of various physical ingredients on inversion.