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?



The total curvature of very knotty knots
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 $(0.264) x (1.658)^E .$
> 
> 
> 


