# Locally flat non-smooth discs

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?

## 1 Answer

You could read the appendix of this paper http://people.mpim-bonn.mpg.de/teichner/Papers/Alexander.pdf

which is very nice. It basically says that for an Alexander polynomial one knot you push a Seifert surface into the 4-ball and then surger it to a disc. The surgery discs need Freedman-Quinn disc embedding though. This cannot be visualised so easily, if at all, since the grope and tower height raising used to construct convergent towers in the proof go over many parallel copies of dual surfaces or grope arms.