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 $\mathbb{Z}$). 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 $\mathbb{Z}⋉\mathbb{Z}\left[1∕2\right]$. These two fundamental groups are known to be the only solvable ribbon groups. Our homological condition implies that the Alexander polynomial equals $\left(t-2\right)\left(t^{-1}-2\right)$ but also contains information about the metabelian cover of the knot complement (since there are many non-slice knots with this Alexander polynomial).