One of the fundamental problems in classical knot theory is to determine the (minimal) crossing number of a given knot. An analogue of the crossing number for knotted surfaces is the minimal number of triple points on projections, called the triple point number. The quandle cocycle invariants have been used to obtain lower bounds for the triple point number.
In [SatShi04], the triple point number of the 
-twist spun trefoil
was determined to be 
using cocycle invariants.
It was the first time that the triple point number was determined
for a specific knot (earlier, only inequalities were known).
See [Hata04,SatShi01b*] for further results on triple point numbers.