Soap Films on Knots

Knotted wires support many interesting soap films. It has recently been proved that every knotted wire supports a soap film that does not touch the entire wire. This goes against the usual mathematical principle that a boundary has no boundary itself. Also, there is a theorem that every smooth embedded closed curve bounds a minimal oriented nonsingular manifold (no triple lines or tetrahedral point singularities). This soap film can be very difficult to imagine. Try it on the trefoil below before looking at the answer.

These images were created with my Surface Evolver program. The Evolver datafiles are available for downloading via the link under each image. (Some of them require Evolver features not available until the release of version 2.10, which should be shortly.)

Trefoil knot

Trefoil Knot

 Trefoil knot

Trefoil Mobius strip Trefoil partially filled
Unorientable nonsingular Mobius band
(knot32m.fe) 		Film partially touching wire
(knot32part.fe)
Trefoil full film Guess before peeking
Film touching entire wire, with triple lines
(knot32f.fe)		 Orientable nonsingular film
(knotrect.fe)
Figure 8 knot

Figure 8 Knot

 Figure 8 knot

Full film Partial film
Full film with tetrahedral point in center.
(fig8full.fe)		 Partial film.
(fig8part.fe)
Unorientable manifold Orientable manifold
Unorientable nonsingular film.
(fig8mob.fe)		 Orientable nonsingular film.
(fig8rect.fe)
Partial film Another partial film
Film with triple line surrounding oval
(oval seen edge on).
(fig8hole.fe)		 Another film with triple line surrounding oval,
with slightly higher energy than the previous film.
(fig8hole.fe)
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