AI 中文总结
本文针对三股辫，利用$SL_2(\boldsymbol{\text{R}})$相关几何，提出构造最优几何辫的方法，得到长度最短测地线的刻画与目标方程，扩展了测地线射影的相关公式。
AI 中文摘要
我们详细描述了从编码编织模式的代数数据构造三股最优几何辫的方法，并提供了计算机代码。我们的最优性准则利用了辫的已知解释：辫是$SL_2(\boldsymbol{\text{R}})$中连接单位矩阵$I$到某个$A \text{∈} SL_2(\boldsymbol{\text{Z}})$的路径的同伦类（相对于端点），即$SL_2(\boldsymbol{\text{R}})/SL_2(\boldsymbol{\text{Z}})$基本群的元素，该商空间等价于经典模曲面$\boldsymbol{\text{H}}/PSL_2(\boldsymbol{\text{Z}})$的单位切丛。主要技术结果是在指定同伦类中找到长度最短的测地线。从另一个独立有趣的角度来看，这相当于在途中经过指定数量的旋转，将庞加莱半平面$\boldsymbol{\text{H}}$的两个给定单位切向量连接起来的最短测地线射影。长度通过一族变形Sasaki度量（有时称为Kaluza-Klein度量）中的黎曼度量测量，其中单位切向量可解释为无穷小转子，称为spinners（旋转器），质量与转动惯量的比值是变形参数。在通用覆盖层$\boldsymbol{\text{SL}}_2(\boldsymbol{\text{R}})$上，该几何是Thurston八种三维模型几何之一。在质量消失极限下，它收敛到更易理解的Carnot-Carathéodory接触几何，其中我们的测地线射影扩展了已知公式。有限质量情况更复杂，需要对目标方程进行数值求解。长度最短测地线的刻画（无重数定理）及由此产生的目标方程是主要原创贡献。论述完整且多维度，面向广泛读者，大量插图是叙述的核心，应查看彩色版本。
英文摘要
We describe in detail and provide computer code for constructing optimal geometric braids of three strands from algebraic data encoding the braiding pattern. Our optimality criterion uses the known interpretation of braids as homotopy classes (rel endpoints) of paths in $SL_2(\mathbb{R})$ joining the identity $I$ to some $A \in SL_2(\mathbb{Z})$, i.e., elements of the fundamental group of $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$, a quotient equivalent to the unit tangent bundle of the classical modular surface $\mathbb{H}/PSL_2(\mathbb{Z})$. The main technical result finds the length minimizing geodesic in a prescribed homotopy class. From another perspective, of independent interest, this amounts to shooting the shortest geodesic that connects, with a prescribed number of spins en route, two given unit tangent vectors to the Poincaré (half-)plane $\mathbb{H}$. The length is measured by a Riemannian metric from a family of deformed Sasaki metrics, sometimes called Kaluza-Klein metrics, whereby unit tangent vectors can be interpreted as infinitesimal rotors, called spinners, and the ratio of the mass to the moment of inertia is the deformation parameter. At the universal covering level $\widetilde{SL}_2(\mathbb{R})$, the geometry is one of Thurston's eight model 3D geometries. In the vanishing mass limit, it converges to the better understood Carnot-Carathéodory contact geometry, where our geodesic shooting extends known formulas. The finite mass case is more delicate, requiring numerical solution of a targeting equation. The characterization of the length minimizing geodesics (No-multiplicity Theorem) and the resulting targeting equation are the main original contribution. The exposition is complete and multi-pronged, aimed at a broad spectrum of readers. (Numerous figures are the backbone of the narrative and should be viewed in color.)
Comments64 pages; 19 figures
