New topologically slice knots

Abstract

In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z semi-direct product Z[1/2]. These two fundamental groups are known to be the only solvable ribbon groups. Our homological condition implies that the Alexander polynomial equals (t-2)(t^{-1}-2) but also contains information about the metabelian cover of the knot complement (since there are many non-slice knots with this Alexander polynomial). Erratum (attached): In Figure 1.5 we gave an incorrect example for Theorem 1.3. We present a correct example.