Groups of knots in homology 3-spheres that are not classical knot groups (Q1074188)

In this paper we attempt to enlarge classical knot groups K by adding a root to a meridian of K. Thus if K is a classical knot group with a meridian \(\mu\), then the groups we study are of the form \(G=K*_{\mu =t^ q}<t>\). This group can always be realized as the group of a knotted 3-sphere in \(S^ 5\). By using explicit geometric constructions we also show that such a group G is a 2-knot group and the group of a knot in a homology 3-sphere. Finally, we show that G is not realizable by any knot in \(S^ 3\).