用强化学习破解复杂纽结，高效找到最少变扣数。
The unknotting number, hard unknot diagrams, and reinforcement learning
- 用强化学习自动寻找纽结变扣序列，降低计算复杂度。
- 确定5.7万纽结的变扣数，发现部分变扣后生成双曲纽结。
- 适合纽结理论研究者，可生成高难度无结图数据集。
我们开发了一种强化学习代理，能经常为最多200个交叉点的纽结图找到最小变扣序列，从而给出变扣数的上界。利用该方法，我们确定了5.7万个纽结的变扣数。通过构造具有相反符号签名的纽结连通和图并叠加其分量，该代理找到了多个例子：在变扣集合中，改变任意一个交叉点都会产生素纽结。基于此，我们证明了在满足某些温和假设下，对纽结$K$和$K'$，存在其连通和的一个图示及$u(K) + u(K')$个变扣，使得改变其中任一交叉点均得素纽结。作为副产品，我们获得了260万条不同的高难度无结图数据；大多数少于35个交叉点。在变扣数可加性假设下，我们确定了43个最多12个交叉点、此前未知变扣数的纽结的变扣数。
原文摘要 · Abstract (English)
We have developed a reinforcement learning agent that often finds a minimal sequence of unknotting crossing changes for a knot diagram with up to 200 crossings, hence giving an upper bound on the unknotting number. We have used this to determine the unknotting number of 57k knots. We took diagrams of connected sums of such knots with oppositely signed signatures, where the summands were overlaid. The agent has found examples where several of the crossing changes in an unknotting collection of crossings result in hyperbolic knots. Based on this, we have shown that, given knots $K$ and $K'$ that satisfy some mild assumptions, there is a diagram of their connected sum and $u(K) + u(K')$ unknotting crossings such that changing any one of them results in a prime knot. As a by-product, we have obtained a dataset of 2.6 million distinct hard unknot diagrams; most of them under 35 crossings. Assuming the additivity of the unknotting number, we have determined the unknotting number of 43 at most 12-crossing knots for which the unknotting number is unknown.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。