  We investigate triangulations of the two-dimensional sphere and torus with
the faces properly colored white and black. We focus on matchings between white
triangles and incident vertices. On the torus our objects are perfect pairings,
whereas on the sphere this is only true after removing one triangle and its
vertices. In the latter case, such matchings (first studied by Tutte) extend
the notion of state in Kauffman's formal knot theory and we show that his Clock
Theorem, in its form due to Gilmer and Litherland, also extends: the set of
matchings naturally forms a distributive lattice. Here the role of state
transposition is played by a simple local operation about black triangles. By
contrast, on the torus, the analogous state transition graph is usually
disconnected: some of its components still form distributive lattices with
global maxima and minima, while other components contain directed cycles and
are without local extrema.
