As a special case, consider a (framed) $\theta$ graph, such that one zipping contains an unknotted component, then the $\theta$ graph is isotopic to a planar one. This follows from a theorem of Scharlemann, which states that if a band sum of knots is trivial, then the knots form a trivial link.

As a special case, consider a (framed) $\theta$ graph, such that one zipping contains an unknotted component and another zipping is a split link, then the $\theta$ graph is isotopic to a planar one. This follows from a theorem of Scharlemann, which states that if a band sum of knots is trivial, then the knots form a trivial link.