Complementary regions for immersions of surfaces

Let F be a closed surface and i:F \to S^3 a generic immersions. Then S^3 - i(F) is a union of connected regions, which may be separated into two sets {U_j} and {V_j} by a checkerboard coloring. For k \geq 0, let a_k, b_k be the number of components U_j, V_j with \chi(U_j) = 1-k, \chi(V_j)=1-k, respectively. Two more integers attached to i are the number N of triple points of i, and \chi=\chi(F). In this work we determine what sets of data ({a_k}, {b_k}, \chi, N) may appear in this way.