卡住的结：刚性、不变量与解开距离
Stuck Knots: Rigidity, Invariants, and Unsticking Distance
AI总结：
本文研究了受刚性约束的结，提出刚性不变量和解开距离，揭示了刚性约束在结简化中的作用。
摘要：
一个卡住的结是一种包含指定交叉点的结图，称为卡住交叉点，其相邻的弦必须保持局部不可分离。这些刚性约束限制了允许的环境同胚，并在研究结嵌入时引入了新的几何特征。在本文中，我们开发了一个针对受此类约束的结的拓扑框架。我们将卡住交叉点建模为空间嵌入中的局部刚性配置，将卡住的结与刚性的空间图论紧密相关，同时保持经典交叉点的上下的信息和方向。我们正式化了相应的同胚概念，并引入了"解开移动"，它释放刚性并允许经典的简化发生。为了检测刚性结构，我们构建了卡住的结的多项式不变量，包括一种HOMFLYPT型不变量和扩展Kauffman括号的状态求和模型。这些不变量表明，即使底层经典结类型保持不变，刚性仍贡献独立的信息。我们进一步引入了一个"放松的同胚"框架，并定义了"解开距离"，这是一个几何度量，量化了将两个卡住的结联系起来所需的最小刚性约束释放数量。这种视角将卡住交叉点解释为同胚的障碍，并突出了释放约束在图示简化中的作用。
英文摘要：
A {\it stuck knot} is a knot diagram containing designated crossings, called {\it stuck crossings}, whose incident strands are required to remain locally non-separable. These rigidity constraints restrict the allowable ambient isotopies and introduce new geometric features into the study of knot embeddings. In this paper we develop a topological framework for knots governed by such constraints. We model stuck crossings as locally rigid configurations in spatial embeddings, placing stuck knots in close relation to rigid spatial graph theory while preserving the classical over-under information and orientation of crossings. We formalize the corresponding notion of isotopy and introduce the {\it unstick move}, which releases rigidity and allows classical simplifications to occur. To detect rigid structure algebraically, we construct polynomial invariants for stuck knots, including a HOMFLYPT-type invariant and a state-sum model extending the Kauffman bracket. These invariants show that rigidity contributes independent information even when the underlying classical knot type remains fixed. We further introduce a {\it relaxed isotopy} framework and define the {\it unsticking distance}, a geometric measure quantifying the minimal number of rigidity constraints that must be released in order to relate two stuck knots. This perspective interprets stuck crossings as barriers to isotopy and highlights the role of constraint release in diagrammatic simplification.
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