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I recently came across this video: Mathematically correct breakfast, which shows how a bagel can be neatly sliced into two identical linked halves.

I'd like to try this with Mathematica. Here's a torus bagel and a Möbius strip cut overlaid. How can I slice the bagel along this cut to create the two halves and pry it open?

With[{opts = {Mesh -> False, Boxed -> False, Axes -> False}}, bagel = ParametricPlot3D[{Cos[t] (3 + Cos[u]), Sin[t] (3 + Cos[u]), Sin[u]}, {t, 0, 2 Pi}, {u, 0, 2 Pi}, opts, PlotStyle -> Directive[Opacity[0.3], FaceForm[Orange], Lighting -> "Neutral"]]; cut = ParametricPlot3D[{Cos[t] (3 + r Cos[t/2]), Sin[t] (3 + r Cos[t/2]), r Sin[t/2]}, {r, -1, 1}, {t, 0, 2 Pi}, opts, PlotStyle -> Directive[Opacity[0.9], FaceForm[Gray], Lighting -> "Neutral"]]; ] bagel ~Show~ cut

Although I've used a Möbius strip for the cut, I have a feeling it is a 2-twist strip because of the two sides — I haven't been able to fit this correctly inside a torus...