TITLE: Measures of entangledness of an open curve
QUESTION [8 upvotes]: 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.

REPLY [3 votes]: 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...