Homomorphisms of Knot Groups on Finite Groups

We describe trial and error computer programs for finding certain homomorphisms of a knot group on a special projective group LF(2, p), p prime, and programs to evaluate #i(3TC ; Z) where 9TC is a finitely sheeted branched covering space of S3 associated with such a homomorphism. These programs have been applied to several collections of examples, in particular to the Kinoshita-Terasaka knots, and we state numerous conjectures based on these experiments. About forty years ago a universal method for obtaining algebraic invariants of knot type was proposed and became standard. The method, as applied to a knot k of type K and having group irK = xi(S3 — k; *), begins with the determination of the homomorphisms of irK on a given group G. These homomorphisms fall into equivalence classes under the action of the automorphisms of G, and a crude preliminary invariant of K is the number of homomorphism classes. In the next stage of the method, we fix a transitive permutation representation of G, perhaps of infinite degree. Each homomorphism class of vK on G is associated with a covering space It of S3 — k such that the number of sheets in the covering is the degree of the permutation representation, and the group H^CM; Z) is an algebraic invariant of the knot type K. In this paper, we shall discuss the means and results of implementing the universal method on a computer when the group G is chosen to be one of the special projective groups Lp = LF(2, p) = PSL(2, p), where p is a prime integer. The universal method has been most thoroughly examined in the case where the group G is cyclic. The determination of the homomorphism classes becomes completely trivial, and all the homology invariants can be deduced from a single matrix, the Alexander matrix. These "cyclic invariants" have been applied with good effect to just about every problem in knot theory, but, alas, when the Alexander polynomial A(x) of the knot reduces to the constant 1 these invariants degenerate and are worthless. It is no good choosing a solvable group for G in such a case; to get useful results from the universal method, we must use nonsolvable groups. Of course, simple groups receive first consideration in this context, and of the families of classical finite simple groups, the family {/_„} is the most manageable. As further encouragement for this study, the group Ls is isomorphic to the alternating group A&, and R. H. Fox has shown in several papers ([5] is a good example) that the homomorphisms of a knot group on Ah have some interesting applications. There are, however, very few clues to suggest what the best method for finding homomorphisms on the special projective groups could be. In addition, almost nothing is known about the homology invariants of homomorphisms on noncyclic groups G. It therefore seems reasonable Received July 17, 1970, revised December 2, 1970. AMS 1969 subject classifications. Primary 5520.