Stokes polyhedra for $X$-shaped polyminos

Abstract : Consider a pair of $\textit{interlacing regular convex polygons}$, each with $2(n + 2)$ vertices, which we will be referring to as $\textit{red}$ and $\textit{black}$ ones. One can place these vertices on the unit circle $|z | = 1$ in the complex plane; the vertices of the red polygon at $\epsilon^{2k}, k = 0, \ldots , 2n − 1$, of the black polygon at $\epsilon^{2k+1}, k = 0, \ldots , 2n − 1$; here $\epsilon = \exp(i \pi /(2n + 2))$. We assign to the vertices of each polygon alternating (within each polygon) signs. Note that all the pairwise intersections of red and black sides are oriented consistently. We declare the corresponding orientation positive.
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Yu. Baryshnikov, L. Hickok, N. Orlow, S. Son. Stokes polyhedra for $X$-shaped polyminos. 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'12), 2012, Montreal, Canada. pp.361-364, ⟨10.46298/dmtcs.3005⟩. ⟨hal-01197226⟩

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