= Solution
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 $g_{ds}(K)=0$.
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