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Let $\gamma$ be a simple (non-self-intersecting) open curve in $\mathbb{R}^3$. I am seeking a measure of its degree of "entangledness," some measure that accords with the intuition one senses with a tangled fishing line. One measure is to connect the two ends of $\gamma$ and use a measure of its degree of knottedness, e.g., its unknotting number, or, perhaps, its writhing number. But it would seem these depend on how the ends are connected, rather than on $\gamma$ alone. Have other natural measures been proposed? I'd appreciate pointers. Thanks!

A year later, I remain interested, especially in some type of energy measure along the lines attempted by Qfwfq. It would be especially pleasing to have a measure that somehow measures the effort it would take to straighten a tangle.

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Consider all rotations of the curve in $\mathbb R^3$ and all linear planar projections. Take the minimum of the number of crossings over all such projections. What do you think of that? – Ryan Budney Jun 1 '13 at 15:13

Peter Roegen works on this problem, with the practical goal of effectively identifying certain knotted proteins. His descriptors (not "invariants", because open curves are topologically unknotted) are Gaussian integrals, and give real-number analogues of finite-type invariants for open curves. The simplest of these is the self-linking number, which is the average (integral) over all rotations of overcrossings minus undercrossings.

To my mind, Gaussian integrals do indeed seem the most natural descriptors for open curves, because they don't artificially close them. They are also quite powerful- they are the practical descriptors of choice for certain classes of knotted proteins. The results and effectiveness of the simplest of these descriptors are summarized in this poster.

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Thanks, Daniel. I too am attracted to not artificially closing the path. – Joseph O'Rourke Jun 1 '13 at 18:20

Ken Millett who I've mentioned in previous answers here has worked on measures of knottedness for open curves. See e.g. this book chapter. The basic idea that he and collaborators have been exploring is to try to define a kind of "dominant knot type" from trying out a large number of ways of closing up the chain.

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Thanks, jc, it does make sense to use statistical properties of all possible closures. In their case, they connect each endpoint by a segment to a point on a large surrounding sphere. – Joseph O'Rourke Jun 1 '13 at 18:18

Perhaps, one could define the "entangledness" of an open curve (of unit lenghth and parametrized in arclength) as the minimum

$$\mathrm{min}_{H}\;E(H), $$

over all possible smooth simple "strightening" homotopies $H:[0,1]\times [0,1]\to\mathbb{R}^3$ with $H(0,s)=\gamma(s)$ and $H(1,s)=s\cdot \gamma\;'(0)+\gamma(0)$ $\forall s\in[0,1]$, of the total work (energy) $E(H)$ needed to strighten up the curve, supposing the curve has an evenly distributed unit mass:

$$E(H)=\frac{1}{2}\int_0^1\int_0^1|\frac{\partial H}{\partial t}(t,s)|^2\;dt\;ds$$

I'm not aware if this idea has been made effective somewhere in the literature...

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This is an intriguing idea, Qfwfq! – Joseph O'Rourke Jun 1 '13 at 18:21
Perhaps, in order to factor out irrelevant movements of the "string", instead of $H(1,s)=(s,0,0)$ one could define $H(1,s):=s\cdot \gamma'(0)+\gamma(0)$, $s\in [0,1]$, otherwise the minimum is not invariant under rigid motions of $\mathbb{R}^3$. I'll edit accordingly. – Qfwfq Jun 1 '13 at 19:58
Yet it's not clear whether strightening w.r.t. the first end needs the same amount of energy as strightening w.r.t. the second end (i.e. minimizing under the condition $H(1,s)=s\cdot \gamma\; ' (1)+\gamma(1)$)... – Qfwfq Jun 1 '13 at 20:03
(...But one could always take the minimum between the two quantities to make the result end-invariant) – Qfwfq Jun 1 '13 at 20:05
Qfwfq: I am not sure that would work either, but another thing to try would be p-energy for other p's. The trouble is, it is conceivable that $inf_H E(H)$ is zero for every open string! One can probably learn quite a bit from Shnirelman's paper (I did not read it closely), maybe his proof works for other p's too and maybe it can be used to establish vanishing, since what Shnirelman is proving is quite a bit stronger: Given a self-diffeomorphism f of a cube, he bounds from above minimal energy of isotopy from f to the the identity. – Misha Jun 6 '13 at 0:01

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