A k-skein is a homeomorph of the 2-vertex graph with k parallel edges. In this paper, the author characterizes the homeomorphs of 3-connected graphs in terms of 3-skeins and draws from this a characterization of 3- connectivity in cubic graphs through 3-skeins. Further investigations of 3-connected graphs yields a characterization of the 3-cuts in such graphs and a sufficient condition for two of these graphs to be isomorphic in terms of the 3-skeins.
Departing from 3-connectivity, the paper also considers the k-cuts in a k-connected graph.