Abstract

We describe an algorithm to find ribbon disks for alternating knots, and the results of a computer implementation of this algorithm. The algorithm is underlain by a slice link obstruction coming from Donaldson’s diagonalization theorem. It successfully finds ribbon disks for slice two-bridge knots and for the connected sum of any alternating knot with its reverse mirror, as well as for 662,903 prime alternating knots of 21 or fewer crossings. We also identify some examples of ribbon alternating knots for which the algorithm fails to find ribbon disks, though a related search identifies all such examples known. Combining these searches with known obstructions, we resolve the sliceness of all but 3276 of the over 1.2 billion prime alternating knots with 21 or fewer crossings.

摘要

我们描述了一种算法，用于查找交替结的带状圆盘，以及该算法的计算机实现结果。该算法的基础是来自唐纳森对角化定理的切片链接障碍。它成功地找到了用于切片双桥结和任何交替结与其反向镜的连接总和的带状盘，以及 662,903 个 21 个或更少交叉点的主要交替结。我们还确定了一些带状交替结的示例，算法无法为其找到带状圆盘，尽管相关搜索确定了所有已知的此类示例。将这些搜索与已知障碍相结合，我们解决了超过 12 亿个交叉点不超过 21 个的素数交替结中除 3276 个以外的所有结点的切片度。