Here's small knot theory question involving a knotted graph.
A theta graph is a $\theta$ shape: two vertices with three parallel edges between them. It has three cycles, each obtained by deleting an edge.
Say we embed a theta graph in $\Bbb R^3$. If each of the three cycles are unknots, is the theta graph then ambiently isotopic to a standard theta graph?
My intuition says yes, but I don't know where to start for proving this. One idea is to fill in each unknot with an embedded disc, but I don't know how to show that this can be done such that the discs don't intersect each other or the other edge. (If they don't intersect each other, we've essentially built a sphere, and we're done.)