从不同视角看，难解的未结通常是容易的
Hard unknots are often easy from a different perspective
- University of Georgia（佐治亚大学）
- RWTH Aachen University（亚琛工业大学）
- Colorado State University（科罗拉多州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对难解未结图，提出ReAPR方法交替进行Pass移动简化与几何重嵌入，成功简化所有已知难解未结及约260万个新示例，总CPU耗时不足30秒。
AI中文摘要：
近期训练AI模型识别纽结的尝试已生成数百万个“难解未结图”，这类图难以通过Reidemeister移动、Pass移动或Reidemeister图上的随机游走简化。但大多数这类图对非图示方法，如纽结补的简化三角剖分（Regina）或纽结群的表示（SnapPy）来说是容易处理的。本文提出ReAPR（重嵌入与Pass重路由）方法，该方法交替进行Pass移动简化与几何重嵌入步骤。重嵌入步骤会在满足交叉约束的前提下，最小化图上高度函数的总变差。我们证明，对于n交叉图，最小总变差为2(n−k)，其中k是为使图成为虚拟交替图所需虚拟化的最少交叉数，这是图的组合不变量。从新视角重投影所得嵌入可揭示此前隐藏的简化操作。ReAPR成功简化了我们所知的所有已发表的难解未结示例，以及几个新的集合（总计约260万个示例），总CPU时间不到30秒。这包括一组以有理缠结形式呈现的Kauffman“挑战”未结，这类未结对非图示方法而言异常困难。
英文摘要：
Recent attempts to train AI models to recognize knots have produced millions of ``hard'' unknot diagrams resistant to simplification by Reidemeister moves, pass moves, or random walks on the Reidemeister graph. Most are easy for the methods based on triangulations of the knot complement found in Regina and SnapPy, but more difficult for simplifiers based on diagrammatic moves. We present ReAPR (Re-embedding And Pass Rerouting), a semi-diagrammatic simplifier which alternates pass-move reduction on a knot diagram with a geometric re-embedding step. The re-embedding minimizes the total variation of a height function on the diagram subject to crossing constraints. We show that for an n-crossing diagram, the minimum total variation is 2(n-k), where k is the least number of crossings one must virtualize to make the diagram virtually alternating; this is a combinatorial invariant of the diagram. Reprojecting the resulting embedding from a new viewpoint reveals previously hidden simplifications. ReAPR successfully simplifies every published hard-unknot example we are aware of, as well as several new collections (~2.6 million examples in total) in under 30 seconds of total CPU time, making it as effective as the best non-diagrammatic unknot recognition method (SnapPy) and about 60x faster. ReAPR is just as effective at simplifying diagrams of random knots, where it is regularly used to simplify diagrams with tens of millions of crossings.
