This article provides initial theoretical results concerning the identifiability of the polytomous local independence model (PoLIM), which is an extension of the basic local independence model (BLIM) to polytomous knowledge structures. It is well-known that the BLIM is not identifiable for graded knowledge structures. This is because there exist parameter transformations, named outcome preserving transformations, that leave unchanged the outcome of the prediction function of the model. In this article a twofold generalization is carried out. On the one side, we extend the notion of gradedness to polytomous structures, and, on the other side, we generalize the outcome preserving transformations to the case of polytomous items. These generalizations lead to the conclusion that the PoLIM is not identifiable for graded polytomous structures. This result generalizes a well-known one with the dichotomous structures. The role of equally informative items in the identifiability of the PoLIM is also investigated. The formal results are accompanied by a numerical example that applies those results to the PoLIM with a concrete polytomous structure that turns out to be graded.