Slinking Slumber (Solution)
by Adam Hesterberg, Anders Kaseorg, and Dan Gulotta
In each level, it's possible for both snakes to eat all the apples and then their own tails, completing the level. For instance, in levels 2 and 7 (which are the same), one possibility is:
The levels don't necessarily have unique solutions, but the linking number of the snakes is unique. In order by level, the linking numbers have absolute values 20, 18, 9, 8, 5, 4, 18, 1, spelling out TRIHEDRA.