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A cubic planar graph gives a cell decomposition of a two-sphere, whose dual complex is a triangulation. If I understand Plateau's laws here, a soap film gives a cell decomposition whose dual complex is a triangulation of a three-sphere. Are there any combinatorial restrictions on the triangulations that can appear, as dual to a genuine soap film in $\mathbf{R}^3$?

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    $\begingroup$ Can you clarify, are you allowing the faces and regions of the soap bubbles to be non-simply connected? For example there are (unstable) toroidal double bubbles satisfying Plateau's laws. One can form a dual complex abstractly, but it might not be a triangulation of S^3 (for example, there may be no vertices). torus.math.uiuc.edu/jms/Images/double $\endgroup$ – Ian Agol Jul 23 '16 at 15:19
  • $\begingroup$ Hi Ian. I'd be just as interested to find out there are combinatorial restrictions on foam with nonsimply-connected bubbles. But the actual question that came up for me is even more restricted: if you draw a cubic planar graph on the boundary of a ball, what kind of minimal surfaces, with Plateau singularities and without interior regions, can you fill inside? $\endgroup$ – David Treumann Jul 23 '16 at 16:14

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