The same results were obtained by T.P. Kirkman for n £ 8. For n = 9 two of the links from which basic polyhedra with 12 crossings could be derived were omitted by Kirkman, but even his incomplete list is sufficient for the derivation of all basic polyhedra for n = 12. Namely, the missing basic polyhedron 12E is the only polyhedron that could be derived from the projection of the link .2 2, denoted by Kirkman as 9Bn, so the uncomplete result of Kirkman (11 of 12 basic polyhedra obtained for n = 12) could be just an omission in the process of derivation. The other more probable reason for this omission is that the polyhedron 12E is the only 2-vertex connected graph, and all the others are 3-vertex connected. The alternating link obtained from the basic polyhedron 12E is equal to the link 11***2. It is its second projection that has no bigons. Hence, some KLs that can be derived from the basic polyhedron 12E can be derived from the basic polyhedron 11***. The complete list of the basic polyhedra with n = 12 crossings was obtained by A. Caudron (1982) by composing hyperbolic tangles, and we will refer to his list and notation.
If we generalize Kirkman's
and introduce not only new triangular, but also p-gonal faces (p
> 3) in the KL diagrams in order to eliminate bigons, we can derive all
basic polyhedra with 6
12 vertices from the
KLs belonging to the L-world or from their direct products.
In this case we may describe the derivation by the corresponding partitions:
where by colored numbers
are denoted the corresponding links p (p
from the L-world. For the larger values of n, the completeness of such
a derivation is an open question.