PreprintKnot groups and slice conditionsJonathan A.HillmanAbstractWe introduce the notions of "\(k\)connectedslice" and "\(\pi_1\)slice", interpolating between "homotopy ribbon" and "slice". We show that every highdimensional knot group \(\pi\) is the group of an \((n1)\)connectedslice \(n\)knot for all \(n\ge 3\). However if \(\pi\) is the group of an \(n\)connectedslice \(n\)knot the augmentation ideal \(I(\pi)\) must have deficiency 1 as a module. If moreover \(n=2\) and \(\pi'\) is finitely generated then \(\pi'\) is free. In this case \(\mathrm{def}(\pi)=1\) also. Keywords: deficiency, knot, ribbon, slice. AMS Subject Classification: Primary 57Q45.
