Graphs

When you consider one part of a design, i.e. a related regular projection, it splits the plane into various zones, the crossings, the edges connecting the vertices (except for the simple circle, i.e. a trivial knot) and zones resembling disks, cells of dimension two. There's one zone that does not resemble a disk - the big zone that extends to infinity. Let's colour this zone in black. Now let's decide at each vertex on the edge of a black cell to extend the colour black to the opposite cell and so forth.

You then get a partition into black and white zones::

The cells of dimension two thus partition themselves into black and white and it is easy to see that it makes a checkerboard where each crossing has the following appearance:

You code this partitioning into a planar graph with signed edges, assigning a vertex to each white cell and drawing an edge between two cells above each crossing which comes between them.

You then put a sign onto this edge depending on the over-under situation: a + sign if the strand which passes over comes from the left when you place it in a white cell, a - sign otherwise. You can represent the sign of an edge graphically with a solid line if + and a dashed line if -.

This data perfectly encodes the regular projection of the interlacing and there is a mapping between the plane projection of the interlacing and this planar graph with signed edges. On the other hand if you place yourself in a three-dimensional sphere (you add a point at infinity to our usual space) and you project onto the sphere (the 'true' two-dimensional cell), the correspondence is not one-to-one. Two planar graphs with signed edges code for the same projection. Which is the second? It's the dual graph that you get by placing the vertices not at the centre of the white cells but of the black cells. They are dual in the sense that ideas of dimension k are replaced by notions of dimension 2-k:

At each edge of one corresponds an edge of the other which cuts it at one single point and its sign is opposite, at each vertex of the one corresponds a face of the other and vice versa. The situation is more symmetrical in the case of the plane, the zone containing the infinity no longer exists and no zone is distinguished from the others, the duality appears.

Reidemeister moves

Les mouvements de Reidemeister moves translate themselves into graphs by these equivalences:

You go from a line to that below by taking the dual graph [I don't understand]. The little feathers on the vertices show that you only consider one part of the graph, that other edges may connect to these vertices and won't be affected by the moves. It is understood that you must consider the same relations when you invert the signs of all the edges. Each Reidemeister move therefore creates four relations on the graphs. There's a one-to-one correspondence between the interlacings and the classes of planar graphs with signed edges, modulo the passage to dual graphs and the mouvements I', II' et III'.

Design

Christian Mercat