Fig. 2.

Correspondence of the polymer networks to n = 0 spin systems. The solid lines denote boundaries between the meaning islands induced by the code on the dual graph (Fig. 1 Right). In the spin model, to each edge a spin S_ij is assigned. Each vertex i contributes to the spin Hamiltonian H_S a factor h_i, which accounts for all possible edge occupancies around this vertex. By the construction h_i (Eq. 5), if a vertex is occupied then at least two of the adjacent edges are occupied. In the present example, a four-junction at vertex 1 (red), which corresponds to a factor a₁b₁₃S₁₃b₁₄S₁₄b₁₆S₁₆b₁₇S₁₇, connects to three linear elements (magenta), e.g., a₇b₇₁S₇₁b₇₉S₇₉, and one three-junction (green), a₃b₃₁S₃₁b₃₂S₃₂b₃₈S₃₈. The corresponding contribution to the spin partition function Z_S is an average over all of the spin orientations. This contribution does not vanish because each spin appears exactly twice in the product because S_ij appears exactly once in both edge configurations of i and j. The weight of this contribution is the product of the b_ij-s and a_i-s for each edge and vertex in the product.