KNOT GROUPS AND SLICE CONDITIONS

Abstract

We introduce the notions of "k-connected-slice" and "π1-slice", interpolating between "homotopy ribbon" and "slice". 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) has deficiency 1 as a module, while (n + 1)-connected-slice n-knots are trivial. If π is the group of a π1-slice 2-knot and π' is finitely generated, then π' is free, and so def(π) = 1.