广义Kirchhoff杆的纽结类型
Knot types of generalized Kirchhoff rods
摘要：
Kirchhoff能量是弧长参数框架曲线空间上的经典泛函，其临界点近似于弹性杆的形状。我们引入了一个广义功能的任意参数化的框架曲线的空间，模型杆轴向拉伸或横截面膨胀。我们的主要结果给出了明确的参数化的所有周期临界框架曲线的广义功能。主要技术工具是闭框架曲线形状相似类的模空间与无限维格拉斯曼流形之间的对应关系。临界框架曲线具有令人惊讶的简单参数化，但它们仍然表现出有趣的拓扑特征。特别是，我们表明，对于每个临界能级有一个参数的框架曲线，其基曲线通过两个环面结类型，呼应了类似的结果Ivey和辛格经典基尔霍夫能源的一个参数家庭。与经典理论不同，广义泛函中存在非环面纽结的临界点。
英文摘要：
Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-sectional inflation. Our main result gives explicit parameterizations for all periodic critical framed curves for this generalized functional. The main technical tool is a correspondence between the moduli space of shape similarity classes of closed framed curves and an infinite-dimensional Grassmann manifold. The critical framed curves have surprisingly simple parameterizations, but they still exhibit interesting topological features. In particular, we show that for each critical energy level there is a one-parameter family of framed curves whose base curves pass through exactly two torus knot types, echoing a similar result of Ivey and Singer for classical Kirchhoff energy. In contrast to the classical theory, the generalized functional has knotted critical points which are not torus knots.
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