There are many knots (e.g. the $P(-3,5,7)$-pretzel knot) that are topologically, but not smoothly slice; "topologically" slice means that there is a locally flat embedding of a disc into the 4-space, such that the disc's boundary is the knot, and smoothly slice means that there is such an embedding that is smooth.
Smooth slice discs are (conjecturally) always ribbon discs, so it is fairly easy to visualise them (see e.g. a ribbon disc for 6_1). Or, using Morse theory, every smooth slice disc may be drawn as a movie, i.e. a sequence of Reidemeister moves and $a$ births, $b$ saddle moves and $c$ deaths from a diagram of the knot to a diagram of the unknot, such that $b = a + c$.
But how can one visualise the disc embedded in a locally flat way which is bounded by a topologically, but not smoothly slice knot? Do such discs e.g. have movies in which some additional non-smooth moves feature?