Primality of multiply connected polyominoes

It is known that the polyomino ideal of simple polyominoes is prime. In this paper, we focus on multiply connected polyominoes, namely polyominoes with holes, and observe that the nonexistence of a certain sequence of inner intervals of the polyomino, called zigzag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to 14, the above condition is also sufficient. Lastly, we present an infinite new class of prime polyomino ideals.