Knot Invariants and Strategic Values in Topological Games: A General Framework and New Theorems

We formulate a general framework for finite, perfect-information, two-player games whose positions are knot or link diagrams and whose terminal payoffs depend on the represented topological objects. The central question is when a knot or link invariant can determine a game's outcome, bound its strategic value, or distinguish positions that are strategically inequivalent. We separate classical knot-theoretic input from game-theoretic conclusions and prove general sufficient conditions based on terminal payoff factorization, invariant-compatible move relations, reachable invariant sets, and minimax recursion. We also give a finite-state quotient theorem and a counterexample showing why a terminal invariant alone generally cannot determine a nonterminal position's value. The results are abstract framework theorems. The paper does not claim that the framework, or the specific conclusions below, exhausts the existing literature on knot games.