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.

我们在实心和理想三叶结的平衡形状（闭合和敞开）之间进行对比研究配置。理想的结点是纯粹的几何抽象，代表了最紧密的配置，并被束缚在一个具有完全柔性，自回避的管中，该管具有不可延伸的中心线和不可变形的横截面。在这里，我们构造了绑在弹性棒中的紧密三叶结的物理实现，并使用X射线断层扫描和3D有限元模拟进行了详细表征。具体来说，我们评估弹性在决定物理结的整体形状，自接触区域，曲率轮廓和截面变形中的作用。我们将弹性结的形状与相应理想构型的先前计算进行了比较。我们关于紧密物理结的结果与纯几何结具有许多相似之处，但也有一些惊人的相似之处，我们将对其进行详细研究。