ALGEBRAIC INVARIANTS OF SIMPLE 4-KNOTS

We propose as an algebraic invariant for a simple 4-knot K with exteri or X the triple (L, eta, [lambda]), where L = Z+pi(2) (X)+pi(3) (X) is a commutative graded ring with unit whose multiplication in positive degrees is determined by Whitehead product, eta is composition with th e Hopf map and [lambda] is the orbit of the homotopy class of the long itude in pi(4)(X) under the group of self homotopy equivalences of the universal covering space X' which induce the identity on L. If K is f ibred these invariants determine the fibre, and the natural Z[t, t(-1) ]-module structures on the homotopy groups capture part of the monodro my. Every such triple with L finitely generated as an abelian group (a nd satisfying the other obviously necessary conditions) may be realize d by some fibred simple l-knot. In certain cases we can show that the triple determines the knot up to a finite ambiguity.