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edited Feb 20 2011 at 18:23 |
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.
Comments
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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edited Feb 20 2011 at 17:54 |
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.
Comments
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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edited Feb 19 2010 at 18:37 |
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.
Comments
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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edited Feb 19 2010 at 4:11 |
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.
Comments
The same proof works for link made out of 3 circles. It is easy capture sphere in a link from 4 circles. (The construction is almost 4 strings go around 4 faces of regular tetrahedron on $S^2$ and link an the same vertexes as in the answer of Anton Geraschenko.Geraschenko) BTW, Can one capture a convex body in a knot?
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edited Jan 11 2010 at 21:58 |
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.
Comments
The same proof works for link made out of 3 circles. It is easy capture sphere in a link from 4 circles. (The construction is almost the same as in the answer of Anton Geraschenko.) BTW, Can one capture a convex body in a knot?
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edited Jan 1 2010 at 21:40 |
My guess is that however you wrap the sphere (without Without loss of generality with 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).
If soOnce it is proved, moving in this direction will eventually allow the sphere to escape. But I have no proof
Proof.I'm making this community wiki in case someone else can elaborate in this direction let $u$ be conformal factor.
EDIT: Note Since Möbius tranform preservs total area $\oint u^2=1$ .
Thus, $\oint u<1$.
It follows that the comments to this post contain for a sutable rotation of $S^2$, we get length decreasing family of Möbius tranforms.
Comments
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edited Dec 15 2009 at 23:37 |
My guess is that however you wrap the sphere (without loss of generality with rope everywhere tangent to the sphere) 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). If so, moving in this direction will eventually allow the sphere to escape. But I have no proof. I'm making this community wiki in case someone else can elaborate in this direction.
EDIT: Note that the comments to this post contain a proof.
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edited Dec 7 2009 at 17:36 |
My guess is that however you wrap the sphere (without loss of generality with rope everywhere tangent to the sphere) 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). If so, moving in this direction will eventually allow the sphere to escape. But I have no proof. I'm making this community wiki in case someone else can elaborate in this direction.
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edited Dec 7 2009 at 17:31 |
My guess is that it's not possible: that, however you wrap the sphere (without loss of generality with rope everywhere tangent to the sphere) some 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). But I have no proof. I'm making this community wiki in case someone else can elaborate in this direction.
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edited Dec 7 2009 at 17:28 |
My guess is that it's not possible: that, however you wrap the sphere (without loss of generality with rope everywhere tangent to the sphere) some 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). It doesn't seem possible that all infinitesimal Möbius transforms can leave the length unchanged, and if there's one that increases the length then reversing the same transform will decrease it. But I have no proof. I'm making this community wiki in case someone else can elaborate in this direction.
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edited Dec 7 2009 at 17:18 |
My guess is that it's not possible: that, however you wrap the sphere (without loss of generality with rope everywhere tangent to the sphere) some 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). It doesn't seem possible that all infinitesimal Möbius transforms can leave the length unchanged, and if there's one that increases the length then reversing the same transform will decrease it. But I have no proof. I'm making this community wiki in case someone else can elaborate in this direction.
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answered Dec 7 2009 at 16:38 |
My guess is that it's not possible: that, however you wrap the sphere (without loss of generality with rope everywhere tangent to the sphere) some 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). But I have no proof. I'm making this community wiki in case someone else can elaborate in this direction.
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