The Homology Characters of the Cyclic Coverings of the Knots of Genus One

Abstract

Let k be a simple closed curve in spherical 3-space 9D1 and let G be its group, i.e., the fundamental group of W, = TZ, k. For each non-negative integer g, denote by G,,) the subgroup of G that consists of those elements whose linking number' with k is divisible by g. Thus G(,) is the commutator subgroup and G(,) G. Denote by DTJ the completion [1] of the covering VJ, of VZ1 that belongs to G(,); this is the gth cyclic covering of 9Z branched about k. If g ? 1 and k is tame TZ1% is an orientable closed 3-dimensional manifold that inherits an orientation from a given orientation of 3-space 9J1. In any case W. is an orientable open 3-dimensional manifold that inherits an orientation from the orientation of 01. The oriented spaces TZM are invariantly associated with the knot type /c of k, by which is meant their topological types are unaltered if k is replaced by p(k) for any orientation-preserving homeomorphism 9 of 9jz1 on itself. Invariants of 9J, are therefore invariants of /c. From now on we assume k to be tame. If g > 1 the relevant invariants of 9DZ are its homology characters: the 1-dimensional torsion numbers aa, . * , mn (arranged in the order in which t-+1 divides r), the first betti number, and the invariants X,(p) of selflinking [2] defined for the odd prime divisors p of zy/m.+1. For g = 0 group G/G(O,) of covering transformations comes into play, and we consider the relation matrix of the 1-dimensional homology group of 9DZ considered as a group with operators, i.e., we consider the Alexander matrix A(t), where t is a generator of G/G(,). The relevant invariants of A(t) are [3] its elementary ideals 0(z(t), d = 1, 2, *--, and the Alexander polynomial A(t) (= g.c.d. U#~)). Seifert [4, 5, 6] has shown that these invariants are all determined by the matrix r = V * I, where V is the 2h x 2h matrix II vi j of crossing numbers of a Seifert diagram of genus h for k, and I is the 2h x 2h block