TITLE: Has anyone tabulated 2-knots? Would anyone like to try?
QUESTION [10 upvotes]: I'd love to have a list of 'small' 2-knots, for some sense of small. It's not clear what one should filter by, but there are two obvious candidates

Write a movie presentation, and count the frames.
Project the 2-knot to R^3, and count the triple points.

Does anyone know if this has been attempted? Such a list could be quite useful.

REPLY [6 votes]: Another complexity filter could be the number of 4-dimensional simplices in a triangulation of the complement.
2-knots complements have simple ideal triangulations.  The proof is very similar to the proof for 1-knots, viewed through the lens of Morse theory on manifolds with boundary.  ie: the Wirtinger presentation is viewed as the fundamental group computation coming from the cell decomposition of a Morse height function (in the stratified sense) on the knot complement. 
It'd be nice if there was a SnapPea-like algorithm for triangulating 2-knot complements.  As far as I know those kinds of details haven't been worked out by anyone.  But it should be reasonably doable.