The KnotLink Game

Abstract

Recently, several new games have been introduced that can be played on knot and link diagrams. One of the first such games, played on knot diagrams, is called the Knotting-Unknotting Game. In this game, one player aims to create an unknot while their opponent tries to produce a nontrivial knot. The Linking-Unlinking Game is similar, but is played on link diagrams. In this game, one player's goal is to produce an unlink while the other player aims to create any nontrivial link. In our paper, we introduce a hybrid of these two games, called the KnotLink game, that can be played on either a knot or a link diagram. Moves and players' goals are similar to those of the previous two games, with one key difference that allows the game board to be transformed from a knot to a link or vice versa during game play. We describe this new game, provide a sample game, and prove several results regarding winning strategies for infinite families of rational knots and links.