Knot Invariants and Strategic Values in Bitopological Games: A General Framework and Applications to Knot and Link Games

Games on knot and link diagrams provide a combinatorial setting in which topological invariants can be used to certify terminal outcomes and to guide winning strategies. Separately, bitopological game theory studies situations in which two topologies are relevant to the same underlying set or function space. This paper formulates a bitopological extension of finite knot-diagram games. A bitopological knot game consists of a finite diagram-state space, two topologies, player-dependent admissibility neighborhoods subordinate to those topologies, a turn rule, a terminal payoff, and a knot or link invariant used to evaluate terminal states. We prove a strategic quotient theorem: when a quotient map preserves terminal payoffs and the sets of successor classes available to each player, the quotient game has exactly the same minimax value and winning regions. We then give a two-topology version and explain how invariant-valued terminal payoffs can reduce a diagram game to a finite invariant game. A finite example shows that changing one player's admissibility system can change the value even when the terminal invariant is unchanged. Applications to the Knotting--Unknotting Game and the KnotLink Game are discussed as extensions, not as claims that the published winning-strategy results for those games have been reproved or superseded. The main contribution is a precise framework and a theorem that can be applied once the admissibility systems and quotient hypotheses are verified for a concrete family of knot or link shadows.