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 $(0.264) x (1.658)^E .$