An Algorithm for Generating a Family of Alternating Knots

Figure 2

An alternating link diagram composed of a set of rectilinear polygons with vertices on the 2-dimensional integer lattice. The polygons are bounded by an axis-aligned square, 𝑆_𝑎, which has side length 𝑎 and has the origin at lower left corner. They are also inscribed in a rectangle, 𝑅_𝑎,𝑏, which has its sides parallel to the diagonals of 𝑆_𝑎, is symmetric with respect to the central point of 𝑆_𝑎, and has sides through the points (0,𝑎−𝑏) and (0,𝑎−𝑏+1). The vertices of the polygons are the lattice nodes lying on the sides of 𝑅_𝑎,𝑏 and they are numbered from 0 to 2𝑎+1.