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One of my favorite theorems is that of Fáry-Milnor, stating that the total curvature of a knot in $\mathbb R^3$ which is not an unknot (an ununknot) is at least $4\pi$.

Can one quantify the way in which knottedness forces an increase in total curvature?

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Doesn't the theorem already quantify it? Milnor shows (Theorem 4.7) that (when all relevant quantities are finite) the minimum curvature in a given isotopy class is $2\pi$ times the "crookedness" (= bridge number). –  Anatoly Preygel Jul 16 '10 at 23:50
    
@Anatoly, right now I do not have access to the paper: I remember vaguely its contents... that's why I was asking :) –  Mariano Suárez-Alvarez Jul 16 '10 at 23:59
    
Ah, fair enough :) –  Anatoly Preygel Jul 17 '10 at 0:07
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It sounds like you're looking for a theorem where "knottedness" is turned from a binary "yes/no" condition to a quantitative condition, like "the figure 8 knot is twice as knotted as the trefoil". Then you want some kind of corrolation between total curvature and this quantitative notion of knotting? The cheap answer would be "take the infima of total curvatures in the knot isotopy class". :) –  Ryan Budney Jul 17 '10 at 0:36
    
Here is a link to Milnor's 1950 paper, "On the total curvature of knots": jstor.org/pss/1969467 . For those without JSTOR access, here is a summary: www-math.mit.edu/~hrm/fas/milnor.pdf . And here is his 1953 paper, "On total curvatures of closed space curves": mscand.dk/article.php?id=1381 . –  Joseph O'Rourke Jul 17 '10 at 1:23

1 Answer 1

up vote 8 down vote accepted

Fields medalist Michael Freedman got involved with knots using a very simple technique, assign a sort of energy integral that becomes infinite if there is a genuine self-crossing, that is if you try to force the curve to change isotopy class. The results relate, at least, to Ryan's comment "the figure 8 knot is twice as knotted as the trefoil".

http://www.jstor.org/pss/2946626

Excerpts from a column by Ian Stewart...the quoting process does not seem to have rendered the mathematical symbols very well, and I do not know what the letter p means, but here is the link: http://www.fortunecity.com/emachines/e11/86/knotprob.html

But it now looks as if the most interesting "energy" concept for knots is not elastic, but electrostatic, as suggested in 1987 by S. Fukuhara of Tsuda College, Tokyo. Imagine the knot to be a flexible wire of fixed length, which can pass through itself if necessary and which has a uniform electrostatic charge along its length. Because like charges repel each other, a knot that is free to move will arrange itself so as to keep neighbouring strands as far apart as possible in order to minimise its electrostatic energy. This minimum energy value is the invariant.

But is it a useful one? Does it have simple, natural properties that mathematicians can exploit? In 1991, Jun O'Hara of Tokyo Metropolitan University proved that the minimum energy of a knot really does increase as the knot becomes more complicated. Only a finite number of topologically different knots exist with energy less than or equal to any chosen value. This means that topological types of knots can be "ordered" in terms of their energy. There is a natural numerical scale of complexity for knots, ranging from simple knots at the low energy end to more complicated ones higher up.

What are the simplest knots? In the most recent issue of the Bulletin of the American Mathematical Society, a team of four topologists - Steve Bryson of NASA's Ames Research Center in California, Michael Freedman and Zhenghan Wang of the University of California at San Diego, and Zheng-Xu He of Princeton University- prove that the simplest knots are exactly what you would expect. They are "round circles"- that is, circles in the everyday sense. Topologists, whose "circles" are usually bent and twisted, have to append an adjective to remind themselves when, as in this case, they are not.

The energy of a round circle is 4, and all other closed loops have higher energy. Any loop with energy less than $6 \pi + 4$ is topologically unknotted - it is a bent circle. More generally, a knot with $c$ crossings in some two-dimensional picture has energy at least $2 \pi c + 4,$ though this bound is probably not the best possible, as the lowest known energy for an overhand knot is about 74. The number of topologically distinct knots of energy $E$ is at most $(0.264) x (1.658)^E .$

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Looking at the original article (ams.org/journals/bull/1993-28-01/S0273-0979-1993-00348-3), $p$ is $\pi = 3.14159...$. Stewart's article (from 1993!) must have been corrupted in the process of putting it on the web. –  Peter Shor Jul 17 '10 at 15:01
    
Thank you, Peter. –  Will Jagy Jul 17 '10 at 17:28

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