A knotted trivalent graph (KTG) is a framed embedding of a trivalent graph in $\mathbb{R}^3$, modulo ambient isotopy. I believe they were first considered (in the quantum topological context, at least) in a paper by D. Thurston. There is an "unzip" move on edges, turning an "H" pattern into a pair of edges (Thanks to Kea for the hand-drawing of this image).

A knotted theta graph unzips in three ways (one unzip for each edge), giving rise to three possible 2-component framed links, related to one another by handleslide (Kirby 2). The following statement looked obvious at first to me, but now I'm beginning to doubt it's even true, and I have no idea how to prove it or to find a counterexample. It `feels' well-known.

Question: Given two KTG's (say knotted theta graphs for simplicity), all of whose unzips coincide (i.e.given an edge e in one, there exists and edge f in the other, such that unzipping along e and along f give ambient isotopic results), does it follow that the KTG's are themselves ambient isotopic? Or do there exist distinct KTG's all of whose unzips coincide?