Instead of selecting achiral KLs from complete lists of KLs with a given number of crossings, we can derive them directly from achiral basic polyhedra. First we select alternating achiral source links derived from achiral basic polyhedra, and then we make tangle replacements preserving achirality.

As the first filter for selecting achiral source KLs the equality of Kauffman polynomials calculated for each source KL and its mirror image is used. After removing all remaining chiral KLs by using the program SnapPea by J. Weeks, actual achiral source KLs are obtained.

There are 9 achiral source KLs derived from the basic polyhedron 6^*: 6^*2:.2, 6^*2.2, 6^*2.2.2 0:2, 6^*2.2:2 0.2 0, 6^*2.2.2 0.2.2.2 0, 6^*2.2:2.2, 6^*2.2.2.2, 6^*2.2.2.2.2 0.2 0, 6^*2.2.2.2.2.2. From 6^*2.2:2.2 are derived two classes of achiral KLs: 6^*p.p:q.q and 6^*p.q:p.q, 6^*2.2.2 0.2.2.2 0 generates 6^*p.p.q 0.r.r.q 0 and 6^*p.q.r 0.p.q.r 0, 6^*2.2.2.2.2.2 generates 6^*p.p.q.r.r.q and 6^*p.q.r.p.q.r, and all other source KLs generate one class each. Instead of the term "family" here we use the general term "class", because p, q, r, ..., are arbitrary tangles, and not only chains of digons.

Hence, from the basic polyhedron 6^* the following classes of achiral alternating KLs are obtained:

From the basic polyhedron 8^* the following classes of achiral alternating KLs are derived: