2-knots with Solvable Group

The 2-knots with torsion-free, elementary amenable knot group and which have not yet been fully classified are fibred, with closed fibre the Hantzsche-Wendt flat 3-manifold HW or a Nil-manifold with base orbifold S(3, 3, 3). We give explicit normal forms for the strict weight orbits of normal generators for the groups of such knots, and determine when the knots are amphicheiral or invertible. The largest class of groups π over which TOP surgery techniques in dimension 4 are known to hold is the class SA obtained from groups of subexponential growth by extensions and increasing unions. No such group has a noncyclic free subgroup. The known 2-knot groups in this class are either torsion-free and solvable or have finite commutator subgroup. (It seems plausible that there may be no others. See Theorem 15.13 of [5] and §4 below for evidence in this direction.) If the group of a nontrivial 2-knot K is torsion-free and elementary amenable then K is either the Fox knot (Example 10 of [2]) or is fibred, with closed fibre R /Z, the Hantzsche-Wendt flat 3-manifold HW = R/G6 or a Nil -manifold. (See Lemma 1.) Each such knot is determined up to Gluck reconstruction, TOP isotopy and change of orientations by its group π and weight orbit (the orbit of a weight element under the action of Aut(π)). This orbit is unique for the Fox knot and for the fibred knots with closed fibre R/Z (the Cappell-Shaneson knots) or a coset space of the Lie group Nil. In each of these cases the questions of amphicheirality, invertibility and reflexivity have been decided, and so such knots may be considered completely classified. (See [5, 6, 7].) We shall give explicit normal forms for the strict weight orbits. Using these, we shall show that (with at most six exceptions) no 2-knot with closed fibre HW is amphicheiral or invertible. The remaining knots have closed fibre the 2-fold branched cover of S, branched over a Montesinos knot k(e, η) = K(0|e; (3, η), (3, 1), (3, 1)), with e even and η = ±1. This include the 2-twist spins of these Montesinos knots, which are strongly +amphicheiral but not invertible, and are reflexive. None of the other knots are amphicheiral or invertible. When the commutator subgroup of a 2-knot group is finite the list of possible groups and weight orbits is known, but the surgery obstruction groups are large, and there are in general infinitely many TOP locally flat knots with a given such group. Thus it is reasonable to restrict attention to those which are fibred. The closed fibre is then a spherical manifold S/π. In this case the question of reflexivity has been settled for 10 of the 17 possible families of such knots [13]. It is likely that none of the remaining knots are reflexive, but this has not yet been confirmed. 1991 Mathematics Subject Classification. 57Q45.