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

