Primality of closed path polyominoes

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.

arXiv commentTo appear in Journal of Algebra and its Applications

References in corpus (3)

• Special simplices and Gorenstein toric rings
• Ideals generated by 2-minors, collection of cells and stack polyominoes
• Primality of multiply connected polyominoes

Cited by in corpus (3)

• Hilbert-Poincaré series and Gorenstein property for some non-simple polyominoes
• Primality of weakly connected collections of cells and weakly closed path polyominoes
• Non-simple polyominoes of Kőnig type and their canonical module