[math-fun] Sullivan spooky knot untangler
The E(K) energy functional is badly behaved for "summed" knots. I.e. tie two (or more) small trefoil knots in some string, then glue the two ends of the sting together. Now the self-energies of the little trefoils will be scale-invariant, but if you tighten them thus obtaining more rope to make the whole large loop of string get greater diameter, then you (energetically) win. Hence, the knots will tighten forever to reduce total energy, and there will be no nice energy minimum. But there is a nice energy min for "prime" knots (this is a published theorem).
Can someone provide a reference (preferably an arxiv link) to the published theorem Warren mentions at the end of his post? Jim Propp On Tuesday, December 31, 2013, Warren D Smith wrote:
The E(K) energy functional is badly behaved for "summed" knots. I.e. tie two (or more) small trefoil knots in some string, then glue the two ends of the sting together. Now the self-energies of the little trefoils will be scale-invariant, but if you tighten them thus obtaining more rope to make the whole large loop of string get greater diameter, then you (energetically) win. Hence, the knots will tighten forever to reduce total energy, and there will be no nice energy minimum.
But there is a nice energy min for "prime" knots (this is a published theorem).
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In the Freedman-He-Wang paper* this theorem is on p. 26: ----- THEOREM 4.3. Let K be an irreducible knot. There exists a simple loop g_K: S^1 -> R^3 with knot-type K such that E(g_K) < E(g) for any other simple loop g: S^1 -> R^3 of the same knot type. ----- --Dan ____________________________________________________ * "Mobius Energy of Knots", Michael H. Freedman, Zheng-Xu He and Zhenghan Wang, Annals of Mathematics, Second Series, Vol. 139, No. 1 (Jan., 1994), pp. 1-50. On 2014-01-01, at 8:47 AM, James Propp wrote:
Can someone provide a reference (preferably an arxiv link) to the published theorem Warren mentions at the end of his post?
On Tuesday, December 31, 2013, Warren D Smith wrote:
The E(K) energy functional is badly behaved for "summed" knots. I.e. tie two (or more) small trefoil knots in some string, then glue the two ends of the sting together. Now the self-energies of the little trefoils will be scale-invariant, but if you tighten them thus obtaining more rope to make the whole large loop of string get greater diameter, then you (energetically) win. Hence, the knots will tighten forever to reduce total energy, and there will be no nice energy minimum.
But there is a nice energy min for "prime" knots (this is a published theorem).
Has anyone been able to locate a working version of John Sullivan's knot-dynamics videos? His homepage http://www.math.uiuc.edu/~jms/ has a link to http://www.math.uiuc.edu/~jms/Videos/ke/ but this URL doesn't work. Jim Propp On Wed, Jan 1, 2014 at 1:50 PM, Dan Asimov <dasimov@earthlink.net> wrote:
In the Freedman-He-Wang paper* this theorem is on p. 26:
----- THEOREM 4.3. Let K be an irreducible knot. There exists a simple loop
g_K: S^1 -> R^3
with knot-type K such that E(g_K) < E(g) for any other simple loop
g: S^1 -> R^3
of the same knot type. -----
--Dan ____________________________________________________ * "Mobius Energy of Knots", Michael H. Freedman, Zheng-Xu He and Zhenghan Wang, Annals of Mathematics, Second Series, Vol. 139, No. 1 (Jan., 1994), pp. 1-50.
On 2014-01-01, at 8:47 AM, James Propp wrote:
Can someone provide a reference (preferably an arxiv link) to the published theorem Warren mentions at the end of his post?
On Tuesday, December 31, 2013, Warren D Smith wrote:
The E(K) energy functional is badly behaved for "summed" knots. I.e. tie two (or more) small trefoil knots in some string, then glue the two ends of the sting together. Now the self-energies of the little trefoils will be scale-invariant, but if you tighten them thus obtaining more rope to make the whole large loop of string get greater diameter, then you (energetically) win. Hence, the knots will tighten forever to reduce total energy, and there will be no nice energy minimum.
But there is a nice energy min for "prime" knots (this is a published theorem).
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Warren D Smith