We study the dynamics of knotted deformable closed chains sedimenting in a viscous fluid. We show experimentally that trefoil and other torus knots often attain a remarkably regular horizontal toroidal structure while sedimenting, with a number of intertwined loops, oscillating periodically around each other. We then recover this motion numerically and find out that it is accompanied by a very slow rotation around the vertical symmetry axis. We analyze the dependence of the characteristic timescales on the chain flexibility and aspect ratio. It is observed in the experiments that this oscillating mode of the dynamics can spontaneously form even when starting from a qualitatively different initial configuration. In numerical simulations, the oscillating modes are usually present as transients or final stages of the evolution, depending on chain aspect ratio and flexibility, and the number of loops.

中文翻译：

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沉积柔性结的周期性运动

我们研究了粘性流体中打结的可变形闭合链沉淀的动力学。我们通过实验表明，三叶结和其他环面结在沉积时通常具有非常规则的水平环形结构，并具有许多相互缠绕的环，彼此之间周期性地振荡。然后，我们通过数值恢复该运动，并发现它伴随着围绕垂直对称轴的非常缓慢的旋转。我们分析了特征时间尺度对链灵活性和纵横比的依赖性。在实验中观察到，即使从定性不同的初始配置开始，动力学的这种振荡模式也可以自发形成。在数值模拟中，振荡模式通常表现为瞬态或演化的最后阶段，