We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if Q is a simple quandle of prime power order then Q and the dual quandle Q* of Q have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles Q. We study a knot invariant based on a quandle homomorphism f : Q1 → Q0. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramin's list. Links to the data computed and various programs in C, GAP and Maple are provided.

中文翻译：

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结和应用的Quandle着色

我们提出了一组 26 个有限 quandles，它们通过着色的数量来区分（直到反转和镜像），所有 2977 个素数定向结最多具有 12 个交叉点。我们还表明，这些结中的 1058 个可以通过来自特定 23 个有限 quandles 的 quandles 的着色数量与它们的镜像区分开来。我们通过 Vendramin 发现的最多 35 个有序的所有 431 个相连的四边形来研究这 2977 个节的颜色。除其他外，我们收集有关所有 2977 节具有相同颜色数量的 quandles 的信息。例如，我们证明如果 Q 是一个素数幂阶的简单四边形，那么 Q 和 Q 的对偶四边形 Q* 对于所有结和猜想具有相同数量的着色，这适用于所有亚历山大四边形 Q。1→ 问0. 我们还应用我们计算的 quadle 着色来获得桥梁指数、中西指数、隧道编号和未打结编号的一些新结果。在附录中，我们讨论了 Vendramin 列表中 quandles 的各种属性。提供了计算数据和 C、GAP 和 Maple 中的各种程序的链接。