  We introduce a topological combinatorial game called the Region Smoothing
Swap Game. The game is played on a game board derived from the connected shadow
of a link diagram on a (possibly non-orientable) surface by smoothing at
crossings. Moves in the game are performed on regions of the diagram and can
switch the direction of certain crossings' smoothing. The players' goals relate
to the connectedness of the diagram produced by game play.
