纠缠环的持久性同调
Persistence Homology Of Entangled Rings
摘要：
聚合物之间的拓扑约束（TC）决定了复杂流体（如乳膏、油和塑料）的行为。日常生活中使用的大多数聚合物溶液都采用线性链;它们的行为被爬行和管理论准确地捕获，这些理论将微观TC与宏观粘弹性联系起来。另一方面，具有非平凡拓扑结构的聚合物，如环，为新技术带来了巨大的希望，但由于它们不遵守标准理论，因此提出了一个具有挑战性的问题;此外，拓扑不变性-即环必须保持不打结和不连接的事实-排除了任何严肃的分析处理。在这里，我们提出了一个明确的，无参数的算法，在聚合物溶液中的TCs，并显示其权力在表征TC的纠缠环。我们分析大规模的分子动力学（MD）模拟通过持久的同源性，一个关键的数学工具，从大型数据集提取强大的拓扑信息。这种方法使我们能够识别环特异性TC，我们称之为“同源螺纹”（H-螺纹），并将它们与聚合物的行为联系起来。它还使我们能够识别，在物理上有吸引力的和明确的方式，尺度依赖的循环，迄今为止，逃避精确的量化。我们发现，虽然线程的邻居慢慢增长的环的长度，随后的TC也广泛的渐近极限。我们提出的方法不限于环状聚合物，可以找到更广泛的应用，在一般的聚合物材料中的TC的研究。
英文摘要：
Topological constraints (TCs) between polymers determine the behaviour of complex fluids such as creams, oils and plastics. Most of the polymer solutions used every day life employ linear chains; their behaviour is accurately captured by the reptation and tube theories which connect microscopic TCs to macroscopic viscoelasticity. On the other hand, polymers with non-trivial topology, such as rings, hold great promise for new technology but pose a challenging problem as they do not obey standard theories; additionally, topological invariance -- i.e. the fact that rings must remain unknotted and unlinked if prepared so -- precludes any serious analytical treatment. Here we propose an unambiguous, parameter-free algorithm to characterise TCs in polymeric solutions and show its power in characterising TCs of entnagled rings. We analyse large-scale molecular dynamics (MD) simulations via persistent homology, a key mathematical tool to extract robust topological information from large datasets. This method allows us to identify ring-specific TCs which we call "homological threadings" (H-threadings) and to connect them to the polymers' behaviour. It also allows us to identify, in a physically appealing and unambiguous way, scale-dependent loops which have eluded precise quantification so far. We discover that while threaded neighbours slowly grow with the rings' length, the ensuing TCs are extensive also in the asymptotic limit. Our proposed method is not restricted to ring polymers and can find broader applications for the study of TCs in generic polymeric materials.
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