# Primality of multiply connected polyominoes

## TL;DR

This paper investigates the primality of polyomino ideals for multiply connected polyominoes with holes, establishing a necessary condition and confirming its sufficiency for polyominoes up to rank 14, and introduces an infinite class of prime ideals.

## Contribution

It introduces a necessary condition for primality in multiply connected polyominoes and proves its sufficiency for polyominoes with rank ≤14, also presenting an infinite class of prime polyomino ideals.

## Key findings

- Necessary condition for primality based on zig-zag walk
- Sufficiency of the condition for polyominoes with rank ≤14
- Identification of an infinite class of prime polyomino ideals

## Abstract

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 non-existence of a certain sequence of inner intervals of the polyomino, called zig-zag 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 class of prime polyomino ideals.