Knot concordance

Two knots are concordant when they cobound a smoothly embedded annulus in $S^3\times [0,1]$. Connected sum makes concordance classes into an abelian group.

A slice knot bounds a smooth properly embedded disk in the four-ball $B^4$.

A slice disk is a smooth proper embedding $D^2\hookrightarrow B^4$ whose boundary is the knot in $S^3=\partial B^4$.

A ribbon disk is a slice disk whose radial Morse function has no interior local maxima. It can be built from disks by attaching bands.

A ribbon knot is the boundary of a ribbon disk.

The slice genus is the smallest genus of a smooth compact connected oriented surface properly embedded in $B^4$ with boundary $K$.

For every unit complex number $\omega$ at which the signature form is nonsingular,

A knot is doubly slice when it is the equatorial cross-section of an unknotted smooth two-sphere in $S^4$.

The double slice genus is the smallest genus of an unknotted closed orientable surface in $S^4$ whose transverse equatorial cross-section is $K$.