Narrow Results

In this paper, we introduce an algebraic structure called a diquandle which is a set equipped with two quandle operations satisfying the right distributive laws. We discuss various examples and some properties of diquandles and also show that a diquandle enables us to distinguish oriented dichromatic links by telling that their coloring sets are different when their arcs are colored by elements of the diquandle.

We introduce and investigate dichromatic singular links. We also construct disingquandles and use them to define counting invariants for unoriented dichromatic singular links. We provide some examples to show that these invariants distinguish some dichromatic singular links.

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 $\le 5$.