So here's a challenge for you; what about
this sequence: 0, 0, 1, 1, 2, 3, 7, 21, 49 - what's
the next number? - (Brady: I'm gonna go 73.)
- No... - (I'm just guessing [laughs])
Of course. And you know
if- any ordinary person would have no
clue, this seems like totally vacuous
and if I tell you it's 165 says, why on
Earth it 165? This is the number of
different mathematical knots with a
given crossing number. Basically it means
there are no mathematical knots with
one crossing or two crossing; there's one
knot with three crossings, there's one
knot is four crossings. There are two knots with
five crossings, there are three knots
with six crossings; and then it starts to
take off and really we get to a lot of
different knots when we have a big
number of crossings. But let's first talk
about mathematical knots; and it's not
maybe what you typically think is a knot.
You see, all kind of things that look oh
yeah yeah yeah these are definitely knots
right? All kind of really beautiful
knots. And of course here, my shoelaces
form what already people would think a
knot. Now, nothing of this is a
mathematical knot because it has loose
ends. If it has loose ends you can in
principle untie it and you can form a
different knot. And so there's nothing
that you can say about these particular
knots because nothing is topologically
locked in. You can always undo the knots
and do something differently. So the
ideal simplest possible mathematical knot
may be just a rubber band, okay?
Well indeed mathematicians call this the
unknot, or the trivial knot, because it's
simply not very interesting. But if you
take this rubber band this is- oh I can
knot it and you try to make some kind of
a knot into it and you stuff it through
itself, and you do weird things; and it may
look very knotted well it's not any
different from what you started with
because I can go and undo the same thing
and it's still the same simple ring. Well
are there any non-trivial mathematical
knots that are not just opening up into
a single loop? And yes there are; for
instance the
simplest knot is the trefoil knot. It's
basically a closed loop as you can see,
in this particular projection we can see
it has three crossings. There's one
crossing here, there's one crossing here,
this one crossing there. As I change this
knot around it may look quite different
and at some point it may have more than
just three crossings, but it never has
fewer than three crossings. It may be
quite irregular, for instance you know
what is this thing? Certainly has a lot
of crossings, but again we're looking for
the minimal number of crossings so we
try to untangle this and open it up as
much as we possibly can. And if you're
patient enough and do that we can see oh,
look, three crossings. There's one crossing
there, there's another crossing there,
there's another crossing down there. And
nothing I can do can get fewer than three
crossings; of course I can twist it, make
a fourth crossing, bend it here and I make
six or seven crossings. But we're
looking for the minimal number and the
minimal number in this case happens to
be three crossings. And with three
crossings, for a mathematician, these are
the same knots;
they're both the trefoil knot. - (Brady: How do you)
(know when you've reached the minimum)
(number of crossings?) - Well you put your
finger on a very sore subject, because
you don't. Now for simple knots there's
no question that these are the
simplest possibilities; but say you're
going to 20, 30 crossings or
something like that - you can never be quite sure.
And as a matter of fact it's still an open
problem to really have a very
complicated Gordian knot and figure out
what type of knot it is. But because of
that, because once you have the simplest
view of a knot in the way that's the one
that has the fewest crossings that tells
you what class you're in. Now if it
happens to be three crossings then
there's only one possible knot, the
trefoil - actually it's not quite true
because if you were to make a mirror
image you could not deform this into its
own mirror image, but mathematicians say
okay look we can mirror this thing; so
we're not counting that separately. Now,
can we make a knot with four crossings?
Here is such a knot. And in this
particular projection you probably just
see 4 crossings; I saw one over here,
there's one over on that side, and then
there's one in here, and there's one down there.
And it turns out- two things: there's only
one type of knot with 4 crossings, and
contrary to the trefoil this actually is
the same as its mirror image. So I can
deform this into its mirror image. As a
matter of fact let's look at that knot
in a little bit more detail; here is the
classical image that you typically see
in a table of knots and that's why I
guess it's called figure eight knot,
Because you can see an obvious figure
eight in there. That's not the only way and
the ow- you see this all the time:
1, 2, 3, 4 crossings. So all of
these are topologically the same thing,
the one that I showed you before in wire
would be shown like this in this
particular diagram. So you can see these
knots all look quite a bit different.
Really they're three-dimensional and so
people have made more three dimensional
renderings; so it's something that's very
close to the first classical image that
we showed, that simple diagram. So this is
your figure eight knot and you can see
it looks yet quite different again - but
in no projection will you ever see fewer
than four crossings. And here we are: this
is the real thing. You know, solid bronze
with a nice green patina done by Steve
Reinmuth in Eugene, Oregon. - (Brady: Can I touch it?)
Absolutely - watch out it's heavy. - (Brady: It's lovely, I love that.)
(Beautiful. Wow, okay that's a fou- what do we)
(call that now? A fou-) - That's a figure eight
knot, and it's a four crossing knot. And we
have seen the trefoil knot, and we have
seen the figure eight knot. Now if you
look at the numbers below here, the first
number here is a three here is a 4, they
indicate what that minimum number of
crossings is; so that's known as the
crossing number. The second number is
simply a serial index that tells you
this is the first of two possible kinds
of knots with five crossings. They cannot
be transformed into one another, they're truly
different. And it has been shown that
there's no third possibility. So if you
have exactly five crossings as a minimal number;
either you have this knot, or you have
that knot, or they're mirror images. As we
go to six crossings we now have three
possible options, and they're all
different and they're just serial number
1, 2, 3 in an arbitrary order. By
the time we go to seven crossings we
have seven different possibilities. So,
that's the first classification of
mathematical knots, you simply try to
figure out what's the minimal number. And
when people first tried to tabulate these
knot tables from a hundred years ago,
there were some tables originally. For a
long time it was believed- I think with
the eight crossing knots that there were
one more, but then actually two were over
overlooked; and others turned out to be
duplicates. So it's really really
challenging. You would think that maybe a
computer program can just take a view of
an arbitrary knot, an arbitrary diagram and tell
you what knot it is? No that's
unfortunately not true. It's true maybe
for the simple knots that have, you know,
maybe up to ten or maybe even twelve
crossings? But then it gets really really
hard because it's so difficult to know
how to deform your knot to: first of all
get to the minimum number of crossings,
and then how to reshape it in that shape
so it looks exactly like something that
you already have in your table. People
have tried to come up with some
mathematical measures to try to describe
knots in in a formal way- - (Brady: Like an)
(algorithm or an equation?) - Absolutely. And
so the the crossing number is certainly
one of those ways of representing a knot.
Another way to represent knots would be
that you're trying- you flatten your knot
as much as much as you can in in the
plane, and like all of these are sort of two
and a half dimensional diagrams which
are planar diagrams basically with
just over/under crossings. So assume
you're on the road and you're traveling
along one of those paths, and every
time you come to crossing you start
numbering them. And you number whether it's
over and under. So on the first
crossing you go under and you call it
number 1. Next crossing you may still
go under - number 2. Then you go over
number 3. Sometimes you come back to
some crossing you've already been - maybe you
have an underpass now you're on the
overpass, okay? So list that again, but
you remember that underpass 3 is the
same as overpass 14. And then overpass
4 is the same
as underpass 22. And that's another way
of describing how that knot is really
linked. People like John Horton Conway, he
tries to chop up these knot diagrams into
smaller pieces that have exactly four
strands coming out of it and then they'll
have a little bit of knottiness inside;
and group them together and link them
together and gradually build a complex
knot out of that. And that led to another,
very formal, description that can
describe any any knot you'd like to
describe. So Mr. Jones has come up with
some kind of a complicated polynomial
that describe knots; there's a Mr. Alexander that
has created an Alexander polynomial. So
you think that, you know, with all this we
really know what's going on;
unfortunately we still do not know
what's going on. If I give you two really
really complicated knots and you- and ask
you are they the same? First you try to
struggle, find out you know what the
crossing number is. Of course if the
crossing number is different you know
they can not possibly be the same. Then
you struggle with the Jones polynomial,
you struggle with the Alexander
polynomial - if any one of these
polynomials is different we know the two
knots are different. And there are two or
three more characteristics of knots. But
even if all the known characteristics of
those two knots are exactly the same we
can still not be sure that they're
indeed the same knot, so there's still an
open challenge to find a true descriptor
that really will will help to make it
quite unique when two knots are
different. - (Brady: So mathematicians still don't)
(fully know what a knot is? They can't fully-)
(they haven't fully got it?) - Well they know
what the knot is, they just don't know if
a particular knot- exactly which label
they should assign to it and whether
two complicated knots are truly
different or not.
[Preview]...here. - (Brady: So this isn't-)
(this is not human DNA then or?) - No let's
assume this is either one of those loops,
or is a circular DNA molecule that could
be a bacterial genome or it could be
just a naturally occurring plasmid.
You are not a knot expert. This is not just-
No I am just a knot-o-phile. - (You're a knot-o-phile!)