通过绳长过滤的雷德迈斯特图的有限纽结理论
Ropelength-Filtered Reidemeister Graphs
AI总结：
该研究利用绳长过滤的雷德迈斯特图为纽结类型构建有限识别框架，证明纽结类型有有限识别长度，通过条件表征有限模式的几何提升，借助相关定理实现组件重建与合并尺度恢复，且结果无需全局投影假设。
摘要：
我们使用绳长过滤的雷德迈斯特图为纽结类型开发了一个有限识别框架。对于固定投影方向，有界绳长粗纽结空间确定一个提升的雷德迈斯特多重图以及经典雷德迈斯特多重图中的单调图图像过滤。此过滤中的有限装饰邻域用作识别证书。我们证明每个有限的雷德迈斯特子多重图在有限绳长级别可见，具有根据交叉复杂度可计算的通用界限。因此，每个纽结类型都有有限的识别长度。我们还通过有限面一致性条件表征了有限模式何时允许顶点相干几何提升。对于严格的绳长子级别，具有均匀可达控制的相对参数平滑定理与多射流横截性一起，通过雷德迈斯特电影实现组件重建。传递到任意小的右松弛可精确恢复几何合并尺度。因此，主要的有限识别和合并尺度结果不需要全局投影——塞尔夫驯服假设。
英文摘要：
We prove relative smoothing with uniform reach control for one-parameter families of positive-reach $C^{1,1}$ knots and combine it with relative multijet transversality to obtain projection-generic regularization inside strict ropelength sublevels. Passing to a lifted Reidemeister multigraph reconstructs the admissible $π_0$ of every strict projection-framed ropelength sublevel, and the right-relaxed diagrammatic merge scale equals the geometric one. The normalised projection-framed sublevels are compact, so connected merge levels are attained. These degree-zero results are organized by a fibre cosheaf: a filtered relational cellular cosheaf of sets on the planar Reidemeister multigraph, whose stalks are strict fibre components and whose costalks are the relations realized by Reidemeister movies. The lifted multigraph is its Grothendieck construction, reconstruction is a descent statement, and the diagram image is its support. The diagram image computes merges exactly when no stalk is persistently separated, and a canonical flat coarsening is the coarsest diagram-indexed model that computes components. For amphichiral knots the spherical diagram image cannot detect mirror barriers, whereas the planar image can; no planar type of the standard diagrams of $4_1$ and $6_3$ is fixed by crossing switch. Symmetric hubs and finite certificates give upper bounds for mirror merges, and sign separators give lower bounds. For finite recognition we distinguish full rooted-ball occurrence, visibility in the monotone diagram-image filtration, and saturation. Every finite Reidemeister submultigraph becomes visible at a finite ropelength level, a visible saturated Barbensi-Celoria characteristic certificate determines the knot type up to mirroring, and a universal computable crossing-complexity bound controls visibility.
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