Finding $k$-cores in graphs is a valuable and effective strategy for extracting dense regions of otherwise sparse graphs. We focus on the important problem of maintaining cores on rapidly changing dynamic graphs, where batches of edge changes need to be processed quickly. Prior batch core algorithms have only addressed half the problem of maintaining cores, the problem of maintaining a core decomposition. This finds vertices that are dense, but not regions; it misses connectivity. To address this, we bring an efficient index from community search into the core domain, the Shell Tree Index. We develop a novel dynamic batch algorithm to maintain it that improves efficiency over processing edge-by-edge. We implement our algorithm and experimentally show that with it core queries can be returned on rapidly changing graphs quickly enough for interactive applications. For 1 million edge batches, on many graphs we run over $100\times$ faster than processing edge-by-edge while remaining under re-computing from scratch.