The topology of poker

We introduce a topological invariant of games, based on homotopy theory, that measures their complexity. We examine it in the context of the “Texas Hold'em” variant of poker, and show that the invariant's value is at least 4. We deduce that evaluating the strength of a pair of cards in Texas Hold'em is an intricate problem, and that even the notion of who is bluffing against whom is ill-defined in some situations. The use of higher topological methods to study intransitivity of multi-player games seems new.