The beauty about knot theory is it's sort
of a special subfield of mathematics
where you don't have to know much
mathematics. You don't need to know a lot
about numbers, you don't need to know
about differential stuff, no integration,
no matrices; it's a branch of mathematics you
can pick up from scratch all by itself
and actually come up with interesting
results and understand what's going on.
People then often ask what about knots?
Is there something practical? I cannot think of
something that's really practical in
terms of shoelaces or so. When they
still believed that our universe is
filled with some ether, ether was that
medium that would transport you know
light waves and so on, and then I think
it was Lord Kelvin and others that
believed that the atoms are just
different knots. So the simplest possible
atoms, say hydrogen, maybe just the unknot.
And then maybe helium is the next more
complicated knot, and so that's the trefoil
knot. And then you go down the
periodic system, and and of course it was
a beautiful and tempting way of bringing
order into the periodic table; except it
was just plain wrong. I mean pretty soon
people found out that is no ether, and of
course also people found that actually
atoms break apart into into particles in
some way, there's more structure to it,
and they're clearly not knots. They're more
more like little planetary systems. And so
that was the end of that. On the other
hand, knots actually now play a role in
advanced physics, in advanced chemistry,
and in advanced biology. You know
physics, you know even with string theory
and and similar things, there- in
statistical mechanics that suddenly they
use some of the things they could prove
with knots that seem to apply there. In
chemistry they're now building molecules
that are deliberately knotted in
complicated ways, there are also people
working with nanoparticles that seem to
find materials that have interesting
properties when, you know, the elementary
elements, at the small scale, are knotted.
And finally in biology you probably have
millions of knots in your body. It turns
out many bacteria, but even in your cell,
some of the DNA strands are actually
closed loops and some of them
actually happened to be knotted. If you
have some very close DNA strand, that may be
knotted, and you can see in this case
there may be a particular group exposed
and it can undergo some chemical
interaction with other molecules. But
then if this thing kind of folds up in a
different way
that part may no longer be available, but
something else may pop out and then that
can interact. So as part of the
regulation of what goes on inside a cell
some of these strands may recoil and
change their shape; and it turns out they
even can change their knottedness,
because there actually are some enzymes
and these enzymes say, can we essentially
come along and snip the strand and
essentially let another strand pass
through and then reconnect it. Something, of
course, mathematicians would never do
because that that's totally against what
you're allowed to do in knot theory. So
that's where knots play a role in very
advanced fields. I have three challenges for
you here. Typical kind of exercises you
want to do when you get started with knot
theory is you look at something that you
might draw arbitrarily, you know having
over-and-under crossings in some way. And
then the question is, what knot is this?
And now you may want to go and take your
pipe cleaners or wires and try to
carefully lay out this particular knot,
with over and under crossings carefully
observed; fuse the ends together. And now
the question is, okay which knot is it?
I'm just counting here, it's going to be
one, two, three, four crossings; so you
think, oh it must be the four crossing knot,
must be the figure eight knot.
True or not true?
Let's take this loop that sort of hangs
down, and just kind of flip it up, right.
So now somehow this thing goes up there,
makes a loop and comes back down and
then goes into this knot. This loop up
there can be untwisted and it goes away,
but then we're just left with this part
with this closing back onto itself. And
we know what that is, that's a trefoil.
So here we see an example where you
have something that looks slightly more
complicated, it's relatively easy to
unravel this and figure out okay this is
just a trefoil knot. Well things can get worse.
What knot is this? Now yeah you can do a
lot of mental exercise. I think I could
take this whole loop and kind of flip it
over and then I can actually pull it
through here. And similarly I can take
this loop here and flip it around the
back to the other side - I'm pretty sure
that this knot will get simpler when you
try to unravel it. I think you should
really lay this out, try it with some
wire or pipe clean and figure out what knot it is.
(Brady: Ah, you're making people do a lot of homework!)
(All right go on!) - Now, more difficult
homework here is yet another knot. One,
two, three, four, five, six: six crossings no
problem. I mean I can try to mentally
flip these loops around or so and
nothing comes of it so I'm probably
stuck. Now I go back to the knot table
that we had before; can we figure out
this six crossing knot - which one of
those three it is? And you may want to do
a lot of deforming and probably will get
very frustrated because you may not be
able to find anything. And he said, wait a
minute is there a fourth different
kind of knot? Not really.
Mathematicians call this a Composite Knot.
It's really two trefoil knots stuck
together. You can see, from here to there
it's clearly a trefoil knot, and we can
take this loop and kind of flip it down.
And then we'd have another trefoil knot
and it's coupled to this one with these
two links. So anytime you can take one
complicated knot and cut it open
somewhere so two strands stick out;
another complicated knot cut it open
somewhere two strands stick out you put it
together, you essentially get a compound knot.
(Brady: And that's not its own knot? It's-)
(that's like)
(doesn't fit into a table of-) - Well it
would make things much much more
complicated. So this allows to simplify
things. Now all of the knots that you see
in here are called prime knots; they
cannot possibly be reduced to something
simpler, they're really on their own as
complicated as they get. But there are
many many complicated knots, that can be
taken apart like a complicated number
can be taken apart into prime factors.
But once you have a prime factor you can
no further simplify it. So all the knot
table does is give you only prime knots,
that are not themselves compound. But now
you can take 7,3 and take- you know 6,3
snip it open here, snip it open there,
stick them together and you make
something that has on the order of 7
plus 6 crossings, but it's not a 13
crossing knot. It's it's this compound knot.
(Brady: You kept that a secret; I feel like you've dropped a)
(bomb on me) - So if you didn't do that, you
know, then the knot table would become
totally unwieldy, okay? Now it would
explode you even more quickly, and even
faster. [Extras] ...you actually are on the other
side. And then you have to go once
through more through the roller-coaster
until you come back to where you started.
So it's just one long string, you know,
essentially tied together...