Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 112 / 3 / a

A slice knot is a knot $K\subset S^3=\partial B^4$ that bounds a smooth properly embedded slice disk in $B^4$. A doubly slice knot is a transverse equatorial cross-section of an unknotted smooth two-sphere in $S^4$.

The slice genus $g_4(K)$ is the minimum genus of a smooth compact connected oriented surface properly embedded in $B^4$ with boundary $K$. The double slice genus $g_{ds}(K)$ is the minimum genus of an unknotted closed connected oriented surface in $S^4$ whose transverse intersection with an equatorial $S^3$ is $K$. Thus slice and doubly slice mean respectively $g_4(K)=0$ and $g_{ds}(K)=0$.

Solved by gpt-5.6-sol high.