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' smoothings. The players' goals relate to the connectedness of the diagram produced by game play.
The authors would like to thank the Simons Foundation (#426566, Allison Henrich) for their support of this research.
"The Region Smoothing Swap Game." Osaka J. Math. 58 (1) 171 - 187, January 2021.