通过斯蒂费尔流形构造的随机纽结
Random Knots via Stiefel manifolds
AI总结：
该研究基于斯蒂费尔流形构造随机棒纽结模型，证明两类投影的纽结类型分布一致，给出六棒情形的纽结分类器，搭建了随机投影到随机纽结拓扑的桥梁。
AI中文摘要：
将固定单纯形随机投影到三维空间并按哈密顿序连接，可生成一种丰富且异常易于处理的随机棒纽结模型。我们证明，高斯投影与哈尔随机斯蒂费尔投影尽管度量形状不同，却具有完全相同的纽结类型分布；且在每个棒数预算下，该模型对恰好能用对应数量棒实现的纽结类型赋予正概率。其线性代数结构可得到精确的边际距离分布、精确的平均平面交叉数及交叉集中性；在首个非平凡的六棒情形中，完整四面体符号模式为每个通用样本提供了精确的未纽结与右旋三叶结分类器。该成果搭建了从随机投影和有限符号几何到随机纽结拓扑的直接桥梁。
英文摘要：
A fixed simplex, randomly projected into three dimensions and joined in Hamiltonian order, produces a rich and unusually tractable model of random stick knots. We prove that Gaussian projections and Haar-random Stiefel projections have exactly the same knot-type law, despite having different metric shapes, and that at every stick budget the model gives positive probability to precisely the knot types realizable with that many sticks. Its linear-algebraic structure yields an exact marginal distance law, an exact mean planar-crossing count, and crossing concentration, while in the first nontrivial six-stick case the complete tetrahedral sign pattern gives an exact unknot-versus-handed-trefoil classifier for every generic sample. The result is a direct bridge from random projections and finite sign geometry to the topology of random knots.
