  We introduce a topological combinatorial game called the Link Smoothing Game.
The game is played on the shadow of a link diagram and legal moves consist of
smoothing precrossings. One player's goal is to keep the diagram connected
while the other player's goal is to disconnect the shadow. We make significant
progress towards a complete classification of link shadows into outcome classes
by capitalizing on the relationship between link shadows and the planar graphs
associated to their checkerboard colorings.
