3.2.7 KLs and mirror curves
Now let us take a look of the classification of mirror curves through knot theory glasses! As mentioned before, every mirror curve can be simply transformed into an interlacing knotwork design, that is, into a projection of some alternating knot. In the history of ornamental art, these curves occur most frequently as knotworks, then as plane curves. Even the name Brahma-mudi (Brahma's knot) denoting Tamil curves refers us to knots. In order to classify these knot projections, we will first transform each one into a reduced (proper) knot projection, i.e., a knot projection without loops, by deleting cells with loops.
In this way, we obtain reduced knot projections with the minimal number of crossings. Two projections or knot diagrams are equal if they are isotopic as graphs, where the isotopy ensures that relations "over"-"under" are preserved. In order to classify our curves, treated as knot projections, we can use the invariant of KL projections defined before.
The rectangular square grid RG[2,2] is the minimal RG from which we can derive some non-trivial alternating KLs (different from the unknot)- the trefoil knot 31 (or 3 in the Conway notation) and the 2-component link 212 (or 2). From RG[3,2] we obtain the knots 74, 62, 31# 31, 51, 52, 41 and 31 (or 3 1 3, 3 1 2, 3#3, 5, 3 2, 2 2 and 3 in the Conway notation), where different mirror-arrangements may give the same projection.
Is it possible to derive every knot projection from some RG with a large enough number of crossings? What is the upper bound for this "large enough"? Which knot projections can be obtained from a particular RG? Which mirror-arrangements in some RG give the same knot projection? Find the minimal RG for a given knot! Can you obtain several non-isomorphic projections of some knot from the same RG? These and many other problems connected with mirror curves represent an open field for research.