Algebraic Invariants of Simple 4-Knots

Abstract

We propose as an algebraic invariant for a simple 4-knot K with exterior X the triple (L, η, [λ]), where L = Z ⊕ π2(X)⊕π3(X) is a commutative graded ring with unit whose multiplication in positive degrees is determined by Whitehead product, η is composition with the Hopf map and [λ] is the orbit of the homotopy class of the longitude in π4(X) under the group of self homotopy equivalences of the universal covering space X′ which induce the identity on L. If K is fibred these invariants determine the fibre, and the natural Z[t,t-1]-module structures on the homotopy groups capture part of the monodromy. Every such triple with L finitely generated as an abelian group (and satisfying the order obviously necessary conditions) may be realized by some fibred simple 4-knot. In certain cases we can show that the triple determines the knot up to a finite ambiguity.