Slice knot

Remarks

The disk $D$ must be properly embedded, meeting $S^3$ only in its boundary. Example: let $K^*$ be the mirror image of $K$ with reversed orientation. Then $K \# K^*$ bounds a ribbon disk, made by spinning a half of $K$ in $B^4$, so it is slice; the square knot is an instance. Non-example: the trefoil knot is not slice, because its signature is $\pm 2$ and the signature of a slice knot is $0$. By Fox and Milnor, if $K$ is slice then $\Delta_K(t) = f(t) f(t^{-1})$ for some $f \in \Z[t, t^{-1}]$, where $\Delta_K$ is the Alexander polynomial. Freedman proved that every knot with $\Delta_K = 1$ is topologically slice. Some of these knots are not smoothly slice, which shows that the two notions differ.