We're going to discuss a little bit
about knots and links,
about how to use coloring knots and
links to study them.
I'll take, say, a piece of string and I'll
project it on a blackboard or a piece of
paper,
that's called the projection of a knot.
And you see when I project it there are
overs and unders
that tell me how it looked when it was
back in three dimensional space.
Now of course you could think of a piece
of string as just going like a line,
but in mathematics it's convenient to
take the two ends of the string
and connect them together so that it
forms a circle all the way around,
right. If you don't stick the two ends
together you can of course always unknot
something just by pulling the strand
through;
and I'm trying to avoid doing that so
that I can talk about when two knots are
different.
It looks like a circle in the sense that
if I had a little ant sitting on this,
for example
here's my little ant, he would walk
around this way and then they keep going
and keep going and keep going and then
he would come back
gradually; he or she - actually it's she
in case of ants -
would come back to where she was at the
very beginning.
And that's what makes it a circle. But
it's a complicated circle in that
it's not just like this, right, this would
be what we call the trivial knot, or the
unknot,
which has no knottings. This is a knot in
the sense that
you can't get from this picture to this
picture without doing something
discontinuous like cutting the knot or
something like that. In mathematics we're
interested in studying
classification of knots; how - what can you
say about knots? How many are there with
with a certain level of of knottings, and
things like that. How do you actually
prove that these two knots are different?
It's not so obvious, in other words you
notice that when you're a kid for
example,
I loved to do it when I was a kid, you
like taking some piece of string,
making it look like it's really
complicatedly knotted and then you fool
the adults by showing that it just yanks
out, right?
So how do you know that isn't true of
this, or that you couldn't get from this
to this
by doing something like making it much
more complicated
and then somehow yanking it out? In fact
we can prove that there's no way to go
from this to this.
The unknot and this knot - this knot is
called the trefoil knot,
it's the first non-trivial knot and how
do you prove that there's no way to go
from this to this,
right? How do you know that if you did a
zillion ways of moving it around
the knottings wouldn't cancel out, right?
(Brady: So just making it)
(is not enough, and like fiddling around)
(with it?) - That's right. You could say, why don't
you just count the number of crossings,
right? The trouble is you can change the
number of crossings in various ways
in a knot. So this knot has three
crossings where there's an over going
on top of an under, there's one over here
there's another one over here,
and there's another one over here. So
there are three places
where one strand lies on top of another
like that. So you could say why don't you
just count the number of crossings? This
one has
zero crossings, this one has three
crossings. And it is true
that the minimum number of crossings you
need for this one is three;
and for this one the number of crossings
is zero. But the trouble is that you can
change the crossing numbers, for example
a simple
dumb bunny way to change the crossing
numbers is introduced into a piece of
a knot.
a pigtail like that; which obviously you
could yank out, right?
There are three basic ways that you can
introduce more crossings in
nota knot. Those are called the Reidemeister
moves. This is the first one, the simplest
one,
and there are two more that involve more
strands. And
the problem is that the Reidemeister
moves seem to increase the number of
crossings, but you can do the opposite
and decrease;
and so you haven't solved the problem of
why can't you get rid of all the
crossings by doing a lot of these Reidemeister moves?
Maybe first complicated and make a
million crossings, but then yank it out
and get down to zero, see what I mean?
(Brady: Is that just like a cosmetic crossing or)
(is it like a lifetime crossing?)
Absolutely, that's the issue. Now let me
draw for you the other Reidemeister moves;
and any motion of a knot can be broken
up into a series of these simple motions.
So this is one, the pigtail, which you
could yank out.
You could yank the pigtail out to just
get a straight line, for example, and a
segment of the knot.
Or you could have two segments of the
knot that are, for example,
not doing anything at all - lying next to
each other. And then you pull
one strand over the other one. So you
would take this, and you could move it
so that it would look for example like
this with one strand
going over and the other one just going
always under;
and obviously you could yank this back
to look like that, right?
Or vice versa. So that's the
Reidemeister2.
And there's one more complicated motion,
R3, which involves three strands.
And there what you do is you have two
strands,
one is crossing over the other, and a
third strand
comes in and then goes, let's say over
both of them say,
like that, so this looks like this and
then what I could do when I have three
strands, for example, I could move it to a
picture where
this guy that's going always over moves
over
further up. Instead of down here he could
move over here over all of them
like that, get what I mean? And now
obviously I could do that by just
pulling the third strand
higher. So that's R3. So those are obvious
simple moves that you can do
and change the projection of a knot. And
now the question is, how do you prove
that there's no way to go from the
Trefoil knot to the unknot,
using these simple moves? Or to give
another example,
there's a simple knot called a figure
eight; and you see it's going to have in
the middle
something that looks like a figure eight,
roughly.
[Music]
This one has zero crossings, this one has
three crossings,
this one - let's count them - has one,
two, three, four crossings. This is a four
crossing knot
and one can prove that actually not only
is there no way to go from this one to
the unknot,
but also these are two different knots.
In other words, just by moving this
around without cutting it,
there's no way to go from the Trefoil
to the figure eight, or the figure eight
to the Trefoil. So
here we have an example, already, right
away of three really different knots. We
can complicate them
by doing these motions, Reidemeister
moves, we could - might try to simplify
them by other Reidemeister moves,
but in fact no matter how many zillions
of moves we make
these three are all different and
there's no way to go from one to the
other.
(Brady: So Professor, it feels like these)
(Reidemeister moves,)
(they're like cosmetic moves, they're like-)
That's right. They don't change the
underlying type of the knot.
They change the way it sits in
three-dimensional space, but they're
equivalent to each other we-
as knots, in the sense that one can be
deformed continuously to the other
without
cutting it. There are tables of knots
which tabulate knots by the minimum
number of crossings.
So this is the only knot with zero
crossings. This is the first non-trivial knot,
the one that's not the unknot, is the
Trefoil knot with three crossings. This is
the one with four crossings, there are
two with five crossings;
and then the number of knots begins to
go up very fast. By the time you're up
into say 12 crossing knots
you've got tens of thousands of them and
so on.
The question is how do you prove that
they're different from each other?
Now, mathematicians have a whole bunch of
different methods of doing that;
using geometry, using algebra, using a
bunch of different ideas;
but there's a very pretty old idea that
can be explained to
you know bright high school students by
using coloring of knots. And that's what
I propose to show you, because it's very
visual,
and you can just see it. So the idea is
the following, I'm going to consider the
following problem.
And for simplicity I'm just going to be
working with three colors.
So what do i mean by three coloring? I
mean that when I take a projection of a
knot, like this
picture over here, I'm going to color
each of the apparent segments that I see
under the projection. So for example in
this case,
because I had four crossings, they look
like there are four
segments over here where it's unbroken.
There's one here and there's another one
that goes all the way around here and
another one here,
and so on. Four crossings gave me four
segments that it's broken up into,
and I'm going to try to color the knot
using three different colors. And what
I'm going to do
is I'm going to try coloring it in a way
so that- with certain rules about how
I'm allowed to color it.
And then I'll see that with these rules,
some knots can be three colored
and some knots cannot be three colored,
and that's going to let me distinguish
one kind of knot from another kind of knot.
So I'll be dividing the knots into two kinds;
those which can be three colored and
those which cannot be three colored.
Now that's not a way of telling all
knots apart, because it's kind of crude
that I'm only dividing them up into two
sets.
But still it'll tell me that some of
them are not equal, cannot be deformed to
the other ones.
Now I have to have a rule because if I
have no rule anything could be colored
with three colors, and then I'm not
gaining any information.
So I have to have a rule. But I want to
have a rule that's compatible with my
Reidemeister moves; in the sense that
if I could color a knot before, there's
an obvious way to then color it after I
do one of these Reidemeister moves.
So when I get to a crossing there's an
over one, and then there are two unders.
One coming in, the other side another one
At a crossing
I will see three colors because I will
have the color of the over,
and then the two unders coming there; so
I have to have three different colors
here, and I have to have some rule
limiting what happens at a crossing. One
obvious rule
to try to use is say, well since that- I'm
trying to do this with three colors
and at a crossing there are three
strands coming together,
the two unders and the over, maybe the
rule should just be:
at a crossing point like this you get
all three different colors. That seems
like the obvious rule to try.
Insist that at the crossing it looks
something like blue over here,
red over here, and say
green. And I do those three colors
together and that would be the rule to
have. That's a natural first guess. It
almost works but not quite. The reason it
doesn't quite work, you can see already,
because it has to be compatible with
these Reidemeister moves that are- as I do
the Reidemeister moves,
the coloration or the color ability is
maintained as I do them.
Now if I do, for example, even just a
pigtail;
so I start off with a single strand like
that and then I take this strand - so this
is somewhere in a big knot diagram,
somewhere in it. And I take this pigtail,
and that of course i can yank out,
and go back to what I had before. So I
can go back and forth
between these two. But notice what
happens that, at the crossing,
I'm coming in from somewh- other stuff
in the knot, so I'm coming in blue,
and I'm going out blue; well that's it,
then I've got to be blue
all the way. So you can see that at this
crossing, that I've introduced with the
pigtail,
in order to make it compatible with what
I had before
I have no choice but to allow at
a crossing
all three colors to be the same at that
crossing. So I'm going to modify
my first guess for the possibility that
all three are the same.
Our tentative new rule is going to be
all three
different, or all three the same, okay?
And that's a rule which we can check now,
case by case, is compatible with these
Reidemeister moves. And now I do the
second Reidemeister move;
and the second Reidemeister move
consisted of pulling one of these over
the other.
I'm coming in from elsewhere in the knot
so I can't change the ends of this,
so I've- I'm now going to have one of
this is coming in blue,
the other one is still coming in red
also, and in fact they've got to go out
the same way to fit with what's going on
elsewhere in the knot;
so here this is going to go out red and
this is still going to go out blue, like
it did before over here.
But what happens in between? Well in
between, this red guy
goes over here and comes over in this
Reidemeister move here, and then goes
back here,
and connects up here - fine. This one is
going to continue
all the way down, and then of course
that's something I could yank out by the
second Reidemeister move.
Just by pulling this back over there and
then I would get my original picture.
But see what happens: remember my rule
was that at the crossing all three are
different or all three of the same;
so with this crossing I have here blue
and red, so now I've got to get green
coming out of there.
I have green coming out of there, coming
down here, and here's green - ah!
And now I'm in luck because there's
green and red and blue coming together,
and that's compatible with my rule
because
at that crossing all three are different.
So we see that this
rule is compatible with the first
Reidemeister move, the second Reidemeister move;
and in fact we could check more cases to
see that it's compatible with the third one.
So this property of being colorable with
this rule
stays the same as I do all three of the
Reidemeister moves - with me?
And, and now I want to have a non-trivial
coloring.
By non-trivial coloring I mean with more
than one color.
So the question I'm going to ask myself
is which knots
have a 3-coloring, with this rule,
with more than one color?
Of course every knot has the silly color,
the trivial coloring, where I color it
all one color - so that's not of interest;
but I want to do it with more than one color.
Okay? And now you'll notice that this one,
the unknot, doesn't- I can only color it
with one color, because there are no
crossings. So this does not have a
non-trivial coloring,
this has only a silly coloring, and as I
move it around that's all I'm going to
keep getting is that silly coloring,
and I'm not going to be able to color it
with different colors. No matter how I
move it I will not be able to color it
non-trivially subject to this rule. Let's
look at the trefoil one, see
is the trefoil colorable? Well okay, so
for example this strand over here
is going to be red. Then there's going to
be another strand which is going to be
blue, let's say this
one. And then there's going to be a
strand that's going to be green,
let's say this one; and now lo and behold
you've noticed that at each of the
crossings we have all three colors.
So this trefoil knot - haha great success,
is in fact three colorable. This is an
example of a three colorable knot.
And this one is not three colorable. And
the last one we're going to consider today
is the figure eight knot. Suppose that
this strand was red,
and then this strand let's say was green,
let's say. Well if this strand is red,
and this one is green, then this one
because it's the third one where they
meet here would have to be blue; so this
one would be blue over here.
Well if this one is blue, and this one is red,
then this one over here also has to be
green because these three are
meeting at this point and that was our
coloring rule - that they either have to be
all three different or the same. But if
this one is green,
and this one is green, then these two
greens meet at this point
so the third one would have to be the
same color - remember all three different
or all three the same -
so we'd have to have green here, but that
contradicts that this guy was red.
So we see that this one is not three
colorable. By this simple technique of studying
three color ability we have proved that
the trefoil
and the figure eight knot are not the
same, because one of them
is three colorable, one of them is not
three colorable.
What we haven't succeeded at proving is
we haven't yet proved that the figure
eight knot
doesn't unknot because all we know
about the figure eight knot is that it's
not three colorable
and that happened also to be true about
the trivial knot. So we haven't yet
proved that these two guys are different.
We've proven that they're all different
from the trefoil, but we haven't yet
succeeded at showing.
But I could generalize this trick in a
slightly more elaborate way
for say five colors, and if I use five
colors this one would turn out to be not
five colorable and this one would turn
out
it is five colorable. And so the upshot
would be that these two
are in fact different. So by doing three
colorability
and adding on a further level of study
to this, the five colorability,
I would indeed have succeeded at proving
that all three of these are different knots.
In particular there's no way to unknot
the trefoil or the figure eight knot
by- without cutting it. And also there's
no way to move as we saw already
the three- the trefoil into the figure
eight; they can't get from one to the other
just by continuous motions. Quite good
for elementary - just,
coloring around, don't you think?
(Brady: Just because this one)
(is three colorable, and that one's - I mean)
(that just seems- and that one's not three colourable-)
Because the point is that after I fill
in the steps that I skipped over,
right, or- which I took an example of but
didn't do the others of-
that the point is that if a knot is
three colorable, as I do these
Reidemeister moves and move it around
and change the number of crossings, the
property of three colorability
is maintained all the way through. For example,
when I did the pigtail and I put in a
pigtail,
it was compatible - in fact I adapted the
rule so as to achieve this property,
and modified my original guess. And so
if this is three colorable, as I move it
around continuously without coloring, without
cutting it; though I change the number of
crossings and may make it like the
kiddies thing,
fantastically more complicated looking
before I try to simplify it,
nevertheless at the very end three
colorability - if it was three colorable
at the beginning
it will be three colorable the end. If it
was not three colorable at the beginning
it will not be three colorable at the end.
And so the fact that the one is three
colorable and the other is not three
colorable, with the given projection,
means that there's no way to move this
to that.
[Preview] And you can understand colorability
better by developing the theory of
polynomial invariance of knots.
Specifically
there's an old polynomial invariant that
goes back about 100 years.
The polynomial invariant for this knot
turns out to be
delta of x - the alexander polynomial - is
x squared minus x plus one.