Constructing Piecewise Linear 2-knot Complements

The groups of high dimensional knots have been characterized by Kervaire [7], but there is still no general description of all 2-knot groups. Kervaire identified a large class of groups that are natural candidates to serve as 2-knot groups and proved that each of these groups is the group of the complement of a smooth 2-sphere in a homotopy 4-sphere. Freedman’s solution to the 4-dimensional Poincaré conjecture implies that the groups Kervaire identified are the groups of locally flat topological 2-knots, but it is not known whether all of them are groups of piecewise linear (PL) 2-knots in S. The best result is due to Levine [9], who observed that the Andrews-Curtis conjecture can be used to show that all the groups identified by Kervaire are groups of PL 2-knots in S. In this note we outline a new proof of Levine’s theorem. The proof given here is entirely 4-dimensional; the Kirby calculus of links is used to give an explicit picture of the 2-knot and its complement. By contrast, the usual proof of Levine’s theorem involves constructing a 5dimensional ball pair whose boundary is the knot. Our proof leads to a piecewise linear knot with one nonlocally flat point. It is clear from the construction that the link of the exceptional vertex is a ribbon link. The proof in this paper is based on a recent construction of Lickorish [10].