The topological framework of circuit topology has recently been introduced to complement knot theory and to help in understanding the physics of molecular folding. Naturally evolved linear molecular chains, such as proteins and nucleic acids, often fold into 3D conformations with critical chain entanglements and local or global structural symmetries stabilised by formation contacts between different parts of the chain. Circuit topology captures the arrangements of intra-chain contacts within a given folded linear chain and allows for the classification and comparison of chains. Contacts keep chain segments in physical proximity and can be either mechanically hard attachments or soft entanglements that constrain a physical chain. Contrary to knot theory, which offers many established knot invariants, circuit invariants are just being developed. Here, we present polynomial invariants that are both efficient and sufficiently powerful to deal with any combination of soft and hard contacts. A computer implementation and table of chains with up to three contacts is also provided.

中文翻译：

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分子电路拓扑的多项式不变量

最近引入了电路拓扑的拓扑框架来补充结理论并帮助理解分子折叠的物理学。自然进化的线性分子链，如蛋白质和核酸，通常折叠成具有关键链缠结和局部或全局结构对称性的 3D 构象，通过链的不同部分之间的形成接触而稳定。电路拓扑捕获给定折叠线性链内链内接触的排列，并允许对链进行分类和比较。接触使链段保持物理接近，可以是机械上的硬连接或约束物理链的软缠结。与提供许多已建立的结不变量的结理论相反，电路不变量才刚刚被开发出来。在这里，我们提出多项式不变量，它们既有效又足够强大，可以处理软接触和硬接触的任何组合。还提供了最多三个触点的计算机实现和链表。