The Celts didn't know about Reidemeister movements, nor about planar graphs with signed edges, but they got along very well! In fact, looking closely at the originals from the middle ages, you can see construction marks which put in place a similar structure; the points which seem to have guided the Irish monks are those which I consider to be the vertices of the graph and of its dual graph. But the motifs which they succeeded in constructing by this method are relatively poor compared to the more ancient motifs carved on menhirs. There are surely methods equally effective to the ones I'm describing, but their secret belongs to the past. After the Irish monks, the technique seems to have been lost for the rare European motifs one can find seem to be copies and juxtapositions of Irish ones. During the Renaissance, Leonardo da Vinci and Dürer made some interlaced designs, calling them humble concatenations, proving that they were based on no general scheme. Few books address the subject and all those that try to describe the methods of construction avoid unifying all the aspects: some treat knots on a square grid, others triangular, others on a circle... However, outside the Western sphere, the Arabs seem to have always mastered the construction of the most sophisticated interlacings, that they used on tiled floors or in illuminated sacred texts. Some of them have the same appearance as Celtic illumination but most are much more formidable. I've searched in vain for a translation of treatises describing their methods. I don't want to underestimate the admirable work of J Romilly Allen or of George Bain, who passed his life collecting Celtic interlacings, compiling the results in a very beautiful book, or even the design manuals of Aidan Meehan, Andy Sloss or Iain Bain (son of George), but their methods only describe certain kinds of knots, especially square borders. However, you will find there Celtic motifs other than interlacings; there are three other types of Celtic design beside knotwork: labyrinths, spirals and animals and I don't know the methods for constructing them.
The method takes the inverse of the construction that associates an interlacing with a planar graph with signed edges; you start with this given and construct the interlacing. All it needs is to place on each signed edge of the design to which it corresponds, a crossing of two threads right in the middle of the edge with the over-under corresponding to its sign. I advise you for this to take a small square of cut out paper and put on it the design corresponding to a + edge and the design corresponding to a - edge and to align this small design parallel to the edge on which you are working and to recopy the design line by line. This appears childish at first sight, but to succeed in designing the image of a crossing by a rotation is not something easy to do.
Following this it's a case of linking up the ends of the strands. Imagine each edge like a wall, with a door right in the middle of the edge. Now stand on the piece of paper, near the door looking down the wall which you are touching, for example, with your left hand. In your right hand you are holding the end of the piece of string which has just passed the crossing through the door in the middle of the edge.
Walk along the wall, as in a maze, keeping your left hand on the wall. When you come to the end of the wall you turn the corner and keep going along the new wall, still touching it with your left hand. At last you come to the door in the middle of this wall and you see where another crossing has taken place. On the ground lie the ends of four strands; join yours to whichever of the four is pointed towards you.
Go through the door, turn right, take the end of the string in your left hand, keep the wall on your right and you're off again! Continue till all the threads are connected.
Christian Mercat 