  Which polyominoes can be folded into a cube, using only creases along edges
of the square lattice underlying the polyomino, with fold angles of $\pm
90^\circ$ and $\pm 180^\circ$, and allowing faces of the cube to be covered
multiple times? Prior results studied tree-shaped polyominoes and polyominoes
with holes and gave partial classifications for these cases.
  We show that there is an algorithm deciding whether a given polyomino can be
folded into a cube. This algorithm essentially amounts to trying all possible
ways of mapping faces of the polyomino to faces of the cube, but (perhaps
surprisingly) checking whether such a mapping corresponds to a valid folding is
equivalent to the unlink recognition problem from topology.
  We also give further results on classes of polyominoes which can or cannot be
folded into cubes. Our results include (1) a full characterisation of all
tree-shaped polyominoes that can be folded into the cube (2) that any
rectangular polyomino which contains only one simple hole (out of five
different types) does not fold into a cube, (3) a complete characterisation
when a rectangular polyomino with two or more unit square holes (but no other
holes) can be folded into a cube, and (4) a sufficient condition when a
simply-connected polyomino can be folded to a cube.
  These results answer several open problems of previous work and close the
cases of tree-shaped polyominoes and rectangular polyominoes with just one
simple hole.
