'Knot divided'. That's a 12-foot tall snow
sculpture; and we did that in
Breckenridge, Colorado. So here is a
Möbius band that makes three twists, okay?
By itself, you know we don't care about
the shape, the Möbius band by itself is
really no different to a mathematician
than the rubber band, it's just a loop,
okay? We're not looking at the
cross-section and whether it's
twisted or not it's just- we're looking
at the centre line of the knot that
defines the knot. But once you take the
Möbius band and you split it down the
middle it becomes a knot. Now it happens
to be a trefoil knot; because now if you
have these strands being nice and soft
you can move them around, and they would
be interlinked in exactly the same way
as this copper structure. If it's only
singly twisted it will simply open up in
a loop that's twice the size and
actually has a full twist in it. So it
depends on how much twist it is; if the
Möbius band is even more twisted, like
five times, you may get this other knot
that we had labelled as 5 sub 1 or
something like that which, you know, gives an
ever more coiling of that same thing. We
started out with a trefoil knot now, and
so here is roughly a trefoil knot. And
then we split this trefoil knot down the
middle, because it happens in this trefoil
knot to be of a Möbius structure. So, let
me back up a little bit and show you two
different trefoils. They're two artistic
trefoil knots; I kind of call them like,
you know: trefoil standing on its own two
feet, because they were carefully
designed so that you know I can actually
stand them up, they're not gonna topple
over. Can you see a difference?
Well I'm claiming one of them is a
Möbius band and the other one is not.
Let's figure out, you know, whether this
one is twisted. So I'm starting on the
side towards the camera up on the lobe
here, and I'm running my finger slowly
down here. I get to the underside of this
foot, I'm coming to the front here, I'm
getting around;
I'm staying here on the upper side of
that foot. I'm coming up here and I'm back to where I
started. On the same side, so this
one is double-sided. Okay I could paint
this side green and I could paint the other
one red and everything would be fine.
But now let's compare that to the other
Möbius band. If I'm starting here again,
on your side, I'm moving down here; I'm
getting to the bottom of this foot, I'm on
the bottom of this highly curved little
thingy, I'm going through the middle
staying at the bottom here, staying at
the bottom side. Coming around, being at
the bottom of this foot. Now I'm coming
up the backside and so basically I'm
coming up to the backside and I've
gotten to the other side. If I continue,
again sliding down here, being on the
upper side, sliding over on the upper
side here, coming back and I'm coming
back to where I started. But I had to go
around the loop twice. - (Brady: So that's Möbius?)
And in that process I would have painted everything
green right? And there's no way I can
paint one of the other sides, without the
two colours running into one another at
some point. So this is the idea of a Möbius
trefoil knot. And then we took a slightly
different shape of- again you can see
this looks roughly like a a trefoil knot,
except it started out with a
cross-section that was more or less, you
know, rectangular originally, but then we
carved out the middle part to make
essentially a split. And because it's a
Möbius band, when I do this I get a knot
that is not really divided. That's why we
call it 'knot divided'. It's like a
rollercoaster where, if you start on this side
of the big lobe, and you go around the
whole rollercoaster; by the time you come-
the first time back up on the mountain
you actually are on the other side.
And then you have to go once through
more through the rollercoaster until
you come back to where you started. So
it's just one long string, you know,
essentially tied together. And while, when
it was still fused without the gap in
the middle, it was a trefoil knot; now
it's a much more complicated knot. How
many crossings does this thing have?
Actually not that easy to see but I
was amazed at MSRI,
you know like a seven-year-old kid;
and I show them that and within seconds
he said twelve. And he realised wherever
we had a crossing in our trefoil knot we
now had split it, we now get four
crossings; we have this situation. And so
indeed he is right, you know, now we have
three times four crossings - it's a 12
crossing knot. So, by the time you're kind
of looking through the knot table you're
already, you know, very far down there
where there's thousands of different
knots, right? And that's just one of them. - (Brady: Are rollercoasters knots?)
I'm pretty sure
that some of them are because, I mean,
that makes kind of fun, you know, you
wanna- many of them have these like
figure eight configuration that go
sometimes around the loop and then they
go through the figure eight; by the time
you go a couple of times through the
middle with figure eight it is very
likely that the things are knotted. But,
you can never be quite sure. There's some
very complicated looking knots that you
can find on the internet, like when you
look for something like difficult unknot
or difficult trivial knot, they show you a
knot that you swear this is a real knot,
but actually it's not.
...experimental part of DNA topology. So the field is called
DNA topology, we study geometry and
topology of circular DNA molecules. Or
linear DNA molecules that locally behave
as circles. - ...same with this one except it
goes back. So starts at the front, curls back,
hold on the way down. Now you're going to
take this lace, put it under that loop...