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We construct a large class of indecomposable positive linear maps from the$2\times 2$ matrix algebra into the $4\times 4$ matrix algebra, which generateexposed extreme rays of the convex cone of all positive maps. We show thatextreme points of the dual faces for separable states arising from these mapsare parametrized by the Riemann sphere, and the convex hulls of the extremepoints arising from a circle parallel to the equator have the exactly sameproperties with the convex hull of the trigonometric moment curve studied fromcombinatorial topology. Any interior points of the dual faces are $2\otimes 4$boundary separable states with full ranks. We exhibit concrete examples of suchstates.

Construction of exposed indecomposable positive linear maps between matrix algebras