Primality of closed path polyominoes

TL;DR

This paper introduces closed path polyominoes, classifies those without zig-zag walks, and proves that absence of zig-zag walks characterizes primality of their associated ideals, supporting a conjecture in polyomino ideal theory.

Contribution

It classifies closed path polyominoes without zig-zag walks and proves this absence is necessary and sufficient for primality, advancing understanding of polyomino ideal primality.

Findings

Classified all closed path polyominoes without zig-zag walks.

Proved that no zig-zag walks is equivalent to primality of the associated ideal.

Provided toric representations for these ideals.

Abstract

In this paper we introduce a new class of polyominoes, called closed paths, and we study the primality of their associated ideal. Inspired by an existing conjecture that characterizes the primality of a polyomino ideal by nonexistence of zig-zag walks, we classify all closed paths which do not contain zig-zag walks, and we give opportune toric representations of the associated ideals. To support the conjecture we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the associated ideal of a closed path. Finally, we present some classes of prime polyominoes viewed as generalizations of closed paths.