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12 Rollback to Revision 10

Without loss of generality we can assume that rope is everywhere tangent to the sphere. Then some infinitesimal Möbius tranform of its surface will shorten the wrapping length while preserving the crossing pattern (and therefore will loosen the rope without causing it to pass through itself). Once it is proved, moving in this direction will eventually allow the sphere to escape.

Proof. let $u$ be conformal factor. Since Möbius tranform preservs total area $\oint u^2=1$ . Thus, $\oint u<1$. It follows that for a sutable rotation of $S^2$, we get length decreasing family of Möbius tranforms.

• The same proof works for link made out of 3 circles.

• It is easy capture sphere in a link from 4 circles. (4 strings go around 4 faces of regular tetrahedron on $S^2$ and link an the vertexes as in the answer of Anton Geraschenko, the first picture)

• BTW, Can one capture a convex body in a knot?

11 deleted 54 characters in body

Without loss of generality we can assume that rope is everywhere tangent to the sphere. Then some infinitesimal Möbius tranform of its surface will shorten the wrapping length while preserving the crossing pattern (and therefore will loosen the rope without causing it to pass through itself). Once it is proved, moving in this direction will eventually allow the sphere to escape.

Proof. let $u$ be conformal factor. Since Möbius tranform preservs total area $\oint u^2=1$ . Thus, $\oint u<1$. It follows that for a sutable rotation of $S^2$, we get length decreasing family of Möbius tranforms.

• The same proof works for link made out of 3 circles.

• It is easy capture sphere in a link from 4 circles. (4 strings go around 4 faces of regular tetrahedron on $S^2$ and link an the vertexes as in the answer of Anton Geraschenko, the first picture)

• BTW, Can one capture a convex body in a knot?

10 added 91 characters in body

Without loss of generality we can assume that rope is everywhere tangent to the sphere. Then some infinitesimal Möbius tranform of its surface will shorten the wrapping length while preserving the crossing pattern (and therefore will loosen the rope without causing it to pass through itself). Once it is proved, moving in this direction will eventually allow the sphere to escape.

Proof. let $u$ be conformal factor. Since Möbius tranform preservs total area $\oint u^2=1$ . Thus, $\oint u<1$. It follows that for a sutable rotation of $S^2$, we get length decreasing family of Möbius tranforms.

• The same proof works for link made out of 3 circles.

• It is easy capture sphere in a link from 4 circles. (4 strings go around 4 faces of regular tetrahedron on $S^2$ and link an the vertexes as in the answer of Anton Geraschenko, the first picture)

• BTW, Can one capture a convex body in a knot?

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6 Just in case people are skimming...