In order to determine argument structure in text, one must understand how individual components of the overall argument are linked. This work presents the first neural network-based approach to link extraction in argument mining. Specifically, we propose a novel architecture that applies Pointer Network sequence-to-sequence attention modeling to structural prediction in discourse parsing tasks. We then develop a joint model that extends this architecture to simultaneously address the link extraction task and the classification of argument components. The proposed joint model achieves state-of-the-art results on two separate evaluation corpora, showing far superior performance than the previously proposed corpus-specific and heavily feature-engineered models. Furthermore, our results demonstrate that jointly optimizing for both tasks is crucial for high performance.