绳长临界性
Ropelength Criticality
摘要：
绳长问题要求在欧氏三维空间中直径为1的打结管的最小长度构形。这种管的核心曲线被称为紧结，其长度是衡量复杂性的结不变量。就芯线而言，厚度约束包括两部分：曲率的上界和自接触条件。我们给出了一组充要条件的临界性相对于这个约束，基于一个版本的库恩-塔克定理，我们在以前的工作中建立。其关键技术难点是光滑摄动下厚度导数的计算。这是通过将厚度写为光滑函数的$C^1$ -紧致族的最小值来实现的，以便应用Clarke定理。我们给出了一些应用程序，包括分类的“超螺旋螺旋”形成的临界曲线没有自接触（仅由曲率约束）和明确的，但令人惊讶的复杂描述的“扣”连接时形成的一根绳子被拉紧另一个。
英文摘要：
The ropelength problem asks for the minimum-length configuration of a knotted diameter-one tube embedded in Euclidean three-space. The core curve of such a tube is called a tight knot, and its length is a knot invariant measuring complexity. In terms of the core curve, the thickness constraint has two parts: an upper bound on curvature and a self-contact condition. We give a set of necessary and sufficient conditions for criticality with respect to this constraint, based on a version of the Kuhn-Tucker theorem that we established in previous work. The key technical difficulty is to compute the derivative of thickness under a smooth perturbation. This is accomplished by writing thickness as the minimum of a $C^1$-compact family of smooth functions in order to apply a theorem of Clarke. We give a number of applications, including a classification of the "supercoiled helices" formed by critical curves with no self-contacts (constrained by curvature alone) and an explicit but surprisingly complicated description of the "clasp" junctions formed when one rope is pulled tight over another.
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