Nonsimple polyominoes and prime ideals (Q281495)

A polyomino \(P\) is a collection of unit squares (also called cells) in the first quadrant of the plane that is connected in the sense that any two cells are connected by a sequence of cells in \(P\) such that adjacent cells share an edge. If \(a\) and \(b\) are two lattice points in the first quadrant, \(a \leq b\) if coordinates of \(a\) are at most coordinates of \(b\). Furthermore \([a,b]\) denote the rectangle with two corners \(a, b\). For any such rectangle \([a,b]\), \(c\) and \(d\) denotes the other two corners.NEWLINENEWLINENow to a polyomino \(P\), one can associate a binomial ideal in a polynomial ring over a field \(k\) in the variables corresponding to the vertices that appear in the polyomino as follows. For every rectangle \([a,b]\) contained in \(P\), one associates the binomial \(x_ax_b - x_cx_d\). Then \(I_P\) denotes the ideal generated by all such monomials over all rectangles that are contained in \(P\).NEWLINENEWLINEA problem of interest is to characterize polyomino \(P\) such that the corresponding binomial ideal \(I_P\) is a prime ideal. In the paper under review, the authors provide a new class of polyominoes whose associated binomial ideals are prime.