Companions of the unknot and width additivity

Knot adjacency and satellites

A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is n-adjacent to the unknot, for some n>0, if and only if it is n-adjacent to the unknot in any companion solid torus. In particular, every model knot of K is n-adjacent to the unknot. Along the way of proving these results, we also show that 2-bridge knots of the form K_{p/q}, where p/q=[2q_1,2q_2] for some integers q_1,q_2, are precisely those knots that have …

解结数、困难解结图与强化学习

The unknotting number, hard unknot diagrams, and reinforcement learning

本文利用强化学习智能体寻找纽结图的最短解结交叉变换序列，确定了大量纽结的解结数，证明了连通和图示中单次解结变换可产生素纽结，并构建了260万困难解结图数据集。

Additivity of Bridge Number of Knots

We provide a new proof of the following results of H. Schubert: If K is a satellite knot with companion J and pattern L that lies in a solid torus T in which it has index k, then the bridge numbers satisfy the following: 1) The bridge number of K is greater than or equal to the product of k and the bridge number of J; 2) If K is a composite knot (this is the case k = 1), then the bridge number of K is one less than the sum of the bridge numbers of J and L.