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 2 𝑎 + 1 .