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 <a href="http://arxiv.org/pdf/math/0311458v2.pdf">paper by D. Thurston</a>. 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).

![Unzip move][1]

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.

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<b>Question</b>: Given two KTG's (say knotted theta graphs for simplicity), all of whose unzips coincide (<i>i.e.</i> 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?
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  [1]: http://1.bp.blogspot.com/_j1PMRiUP1iY/RgCC3zmPPQI/AAAAAAAAAGI/lRg-3a_fm90/s320/cubeUnzi.JPG