A diquandle is a set equipped with two quandle operations interacting via a kind of distributive laws which come from Reidemeister moves on dichromatic links. This algebraic systems provide coloring invariants for dichromatic links. In this paper, we give explicit constructions of free diquandles and diquandle presentations, and then discuss Tietze transformations for the diquandle presentations. We also introduce the fundamental diquandles for dichromatic links. Particularly, we describe the fundamental diquandles and diquandle counting invariants for knots and links in the solid torus via annulus diagrams. We append the tables of diquandles and dikei’s of orders $\le 5$.

二分法是一个配备有两个四分法运算的集合，它们通过一种分配律相互作用，该分配律来自二色链接上的雷德迈斯特移动。该代数系统为二色链接提供了着色不变量。在本文中，我们给出了自由二分位数和二分位数表示的显式构造，然后讨论了二分位数表示的 Tietze 变换。我们还介绍了二色链接的基本二象限。特别地，我们通过环图描述了实体环面中的结和链接的基本二分位数和二分位数计数不变量。我们附加了 diquandles 和 dikei 的阶数表$\le 5$。