Let us now describe four general rules for combining plate designs and/or mirror curves. The first three of these rules are given by P. Gerdes (1999), and the fourth is proposed by S. Jablan. We will restrict our consideration to mirror curves placed in polyominoes with square cells.
The first rule defines a combination of two mirror curves that share one edge of an open cell on their borders. Such a composition corresponds to the direct product of KLs, and it was probably one of the most exploited constructions in knotwork art. For given mirror curves M1 and M2, this kind of direct product we will call ×-direct product and denote it by M1×M2. If we combine two mirror curves in this way, first with c1, and the other with c2 components, the result is a new mirror curve with c1+c2-1 components. Hence, the ×-direct product of two 1-component mirror curves is a new 1-component mirror curve. This idea was used, for example, in the Tchokwe designs and in many Celtic friezes.
As a particular application of the first rule, we can add a single square to the border of any monolinear mirror curve. This transformation is the addition of an external loop. It does not change the number of components and can be repeated as many times you like. For example, the Tamil (unknot) design is created by a series of external loop additions, beginning from the RG[1,1]; the other knot design by adding loops to the RG[4,3]; and the third knot design by adding loops to the RG[5,3]. The same construction is used for Tchokwe designs. In knotwork art it has a decorative function.
The second rule is exactly the one by which we define the direct product K1#K2 of two knots K1 and K2 in knot theory. In the language of mirror curves M1 and M2, it means that we cut one external edge of each mirror-curve M1 and M2, and reconnect them again to obtain a new mirror-curve, that will be denoted by M1||M2.
The third rule is restricted to plate designs: two monolinear plate designs whose overlapping contains exactly two cells will give a new monolinear plate design. The schematic interpretation of the third rule is given in the corresponding figure.
In order to introduce the fourth rule we need a new operation, addition. The addition of a plate design P1 to plate design P2 is an edge-to-edge identification of their border cells belonging to rectilinear borders. In the same way, we can add a plate design P1 to some mirror curve M placed in some polyomino.
The fourth rule is: an RG[a,b] for which b|a, added to any monolinear mirror curve M (or monolinear plate design P2) incident to it along the edge b, will give a monolinear design. In particular, any square RG added to a monolinear design gives a new monolinear design.
Rule 4 can be also applied to mirror curves. In this case, we can add a mirror curve M2 to a monolinear mirror curve M1, that every point of an edge b of the polyomino in which M2 is placed belongs to a different component of M2. The new mirror curve M1+M2 will be monolinear.