If PL(v,z) = Σbi(v)zi is the skein polynomial of a link L, and D = D1 * D2 is the diagram which is a planar star (Murasugi) product of D1 and D2 then bϕ(D)(v) = bϕ(D1)·bϕ(D2)(v) where ϕ(D) = n(D)– (s(D) – 1) and n(D) denotes the number of crossings of D, and s(D) is the number of Seifert circles of D.

Let L be an alternating link and be its reduced (or proper) alternating diagram. Let w() denote the writhe of [3], i.e. the number of positive crossings minus the number of negative crossings. Let VL(t) be the Jones polynomial of L [2]. Let dmaxVL(t) and dminVL(t) denote the maximal and minimal degrees of VL(t), respectively. Furthermore, let σ(L) be the signature of L [5].

A conjecture of Fox about the coefficients of the Alexander polynomial of an alternating knot is proved for alternating algebraic (or arborescent) knots, which include two-bridge knots.

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.

Let K be a knot in a manifold M. Corresponding to a representation of Π1(M — K) into a transitive group of permutations there is a branched covering space of M. K is covered by which may be a link of several components. The set of linking numbers between the various components of has long been recognised as a useful knot invariant. Bankwitz and Schumann used this invariant in considering dihedral coverings of Viergeflechte.

In this paper, we will prove, as a consequence of the main theorem,

THEOREM A. (See Corollary 2.6). The group of an alternating knot, for which the leading coefficient of the knot polynomial is a prime power, is residually finite and solvable.