Knot Groups and Slice Conditions

We introduce the notions of “k-connected-slice” and “π1-slice”, interpolating between “homotopy ribbon” and “slice”. We show that every high-dimensional knot group π is the group of an (n− 1)-connected-slice n-knot for all n ≥ 3. However if π is the group of an n-connected-slice n-knot the augmentation ideal I(π) must have deficiency 1 as a module. If moreover n = 2 and π is finitely generated then π is free. In this case def(π) = 1 also. An n-knot is a locally flat embedding K : S → S. Such a knot K is homotopy ribbon if it is a slice knot with a slice disc whose exterior W has a handlebody decomposition consisting of 0-, 1and 2-handles. The dual decomposition of W relative to ∂W has only (n+1)-, (n+ 2)and (n+3)-handles, and so the inclusion of ∂W into W is n-connected. More generally, we shall say that K is k-connectedslice if there is a slice disc with exterior W such that (W, ∂W ) is kconnected, and that K is π1-slice if the inclusion of the knot exterior X(K) = Sn+2 −K(Sn)×D2 into the exterior of some slice disc induces an isomorphism on fundamental groups. Every ribbon knot is homotopy ribbon [4], while if n ≥ 2 “homotopy ribbon” ⇒ “n-connected-slice” ⇒ “π1-slice” ⇒ “slice”. Nontrivial classical knots are never π1-slice, since the longitude of a slice knot is nullhomotopic in the exterior of a slice disc. (A 1-knot is “homotopically ribbon” in the sense used in Problem 4.22 of [6] if and only if it is 1-connected-slice.) It is an open question whether every classical slice knot is ribbon. However in higher dimensions these notions are generally distinct. Every even-dimensional knot is slice, but a knot group is the group of a ribbon n-knot (for n ≥ 2) if and only if it has a Wirtinger presentation of deficiency 1 [11]. (More generally, if W is homotopy equivalent to a finite 2-complex and χ(W ) = 0 then def(π1(W )) ≥ 1.) There are n-knot groups with deficiency ≤ 0 for every n ≥ 2. In this note we shall show that every high-dimensional knot group π is the group of an (n−1)-connected-slice n-knot for all n ≥ 3. However the groups of n-connected-slice n-knots satisfy constraints related to 1991 Mathematics Subject Classification. 57Q45.