Summary: Given an immersion \(\varphi : \mathbb P^1 \to \mathbb P^2\), we give new approaches to determining the splitting of the pullback of the cotangent bundle. We also give new bounds on the splitting type for immersions which factor as \(\varphi : \mathbb P^1 \cong = D \subset X \to \mathbb P^2\), where \(X \to \mathbb P^2\) is obtained by blowing up \(r\) distinct points \(p_i \in \mathbb P^2\). As applications in the case that the points \(p_i\) are generic, we give a complete determination of the splitting types for such immersions when \(r \leq 7\). The case that \(D^2 = -1\) is of particular interest. For \(r \leq 8\) generic points, it is known that there are only finitely many inequivalent \(\varphi\) with \(D^2 = -1\), and all of them have balanced splitting. However, for \(r = 9\) generic points we show that there are infinitely many inequivalent \(\varphi\) with \(D^2 = -1\) having unbalanced splitting (only two such examples were known previously). We show that these new examples are related to a semi-adjoint formula which we conjecture accounts for all occurrences of unbalanced splitting when \(D^2 = -1\) in the case of \(r = 9\) generic points \(p_i\). In the last section we apply such results to the study of the resolution of fat point schemes.