Knot groups and slice conditions

Jonathan A.Hillman

Abstract

We introduce the notions of "\(k\)-connected-slice" and "\(\pi_1\)-slice", interpolating between "homotopy ribbon" and "slice". We show that every high-dimensional knot group \(\pi\) is the group of an \((n-1)\)-connected-slice \(n\)-knot for all \(n\ge 3\). However if \(\pi\) is the group of an \(n\)-connected-slice \(n\)-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.