Splittings of Tangles and Spatial Graphs

Menasco proved the surprising result that if $G$ is a reduced, alternating, connected projection of a link $L$ and $G$ is prime then $L$ is prime. This result has been generalized to other classes of links, tangles, and spatial graphs. We draw attention to some issues with previous splitting results about tangles and spatial graphs, and obtain new more general results for tangles and spatial graphs.