Thilde Kamstrup-Jespersen
AkMat Blok 1, efterår 2008
Knots
Introduction
Over 100 years old field, but some of the most exiting results just came up 15 years ago, and the theory
is still evolving. It can be explained at an elementary level and normal people understands it because
many of the unsolved problems are easily stated. Some of the open problems are problems that nonexperts can attack and maybe solve.
What is a knot? It is a smooth image of a circle, a closed curve, in 3-dimensional space that does not
intersect itself. It is a knottled loop of string with no loose ends, it is truly knotted and you can not
untangle it unless you use a scissors (and that’s not allowed.
The unknot/trivial knot (The unknotted circle, no crossings – the places where the knot crosses itself),
the trefoil knot and the figure-eight knot are the most basic knots.
History:
Knottheory was motivated by chemistry.
In the 1880s a chemist called Lord Kelvin believed that there was a connection between knots
and atoms. He claimed that atoms are knotted vortices in the ether.
Peter Guthrie Tait was convinced by the theory and tried to list all the possible knots, so he
could create a table of the elements in the field of chemistry.
He spent lots of years tabulating knots. Later they found out that Lord Kelvin was wrong,
because a better model of atomic structure appeared and chemists lost interest in knot-theory.
But now mathematicians got interested with knots; How to tabulate knot? Are two knots the
same?
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Thilde Kamstrup-Jespersen
AkMat Blok 1, efterår 2008
Tabulating Knots
Table of knots:
These knots are all distinct and you are not able to deform one of them to look like one of the others.
But you can change the form of a knot. That is called the projection of the knot. So projections are
different pictures/forms of the same knot
F.x. in the picture below you se the trefoil knot, even though it doesn’t look like it. It looks like it has
more crossings, but when you try to untangle it you get the trefoilknot.
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Thilde Kamstrup-Jespersen
AkMat Blok 1, efterår 2008
But how to come from one projection to another projection of the same knot?
There are some defined ways:
- Ambient isotopy: Rearranging of the string (of course you’re not allowed to shrink a part of
the knot down to one point to get rid of the knot – like when you tie your
shoes).
- Planar isotopy: When we only deform the knot within the projection plane.
- Reidemeister-moves: One of the ways to change a projection that will change the relation
between crossings, but not the knot represented by the projections.
There are 3 Reidemeister-moves:
1. put in or take out a twist in the knot
2. add two crossings or remove two crossings
3. slide a strand of the knot from one side of a crossing to the other side of the crossing.
Along with the projections a problem occurs:
“if we have the projection of a knot – is it the unknot?”
1961 Wolfgang Haken made a foolproof procedure that could deal with this. It was an algorithm for a
computer-program and it could tell whether a knot that is given is the unknot. The algorithm is still too
complicated today, nobody has written the program yet.
(Haken was, by the way, the first to try solving the Four-Color Theorem. It was the first
computerbased proof of a major mathematical theorem. He was known in the computer-field and he
knew what he was doing.)
Knot composition and decomposition
We can multiply knots by taking 2 projections of nontrivial knots J and K (these are the factor knots).
We then compose them (multiply them) by cutting the string in J and K and tie the loose ends together
(of course it has to be two ends so they don’t make a new unwanted crossing.)
The result is a new knot: the composite knot J#K.
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Thilde Kamstrup-Jespersen
AkMat Blok 1, efterår 2008
Here we can draw an analogy to the positive integers: An integer is composite if it is the product of
two positive integers (where neither of them is 1). And also: if you multiply a positive integer with 1
you get the same positive integer. The same goes for the knots: If K is a nontrivial knot and U is the
trivial knot, then K#U = K.
Question: “Given a knot K, does there exist a knot L, so that K#L is the unknot?”
NO, because there are no way to take the composition of two nontrivial knots and get the unknot!
Again we know that the integer 1 is not the product of two positive integers, each greater than 1.
The unknot is not composite, because then every knot would be composite. And since every knot is the
composition of itself with the unknot, then every knot would be the composition of itself with the
untrivial factor knots that made up the unknot.
A knot that is not the composition of two other nontrivial knots is called a prime knot. And as an
integer factors into a unique set of primes, then a composite knot factors into a unique set of prime
knots.
One way where the knots differs from the positive integers, because there is only one way to multiply
two integers, but more than one way to take the composition of two knots. There is a choice of where
to remove the arc that will affect the outcome.
Fundamental problems
How many knots are there? The answer to that is “Infinitely many”. Well: How do they look? How do
we tabulate them?
In the 1880s the first (successful) work on tabulating knots was made by Peter Guthrie Tait. He listed
all the alternating knots up to 10 crossings.
(Alternating knot: a knot with a projection that has crossings that alternate between over and under
as you go one direction in the knot.)
In 1899: C.N. Little published a table of 43 nonalternating knots up to 10 crossings.
– BUT after 75 years Kenneth A. Perko proved that 2 of them where the same. They where projections
of the same knot.
This leads us to the first main unsolved problem in knottheory:
Are two given knots the same or different? K = L ?
In 1927 the first proof came up that the knots of up to 9 crossings were actually distinct. This proof
was based on using polynomials applied to knots.
In 1982 Thistlethwaite made a computer-program that listed the prime knots up to 13 crossings:
Number of crossings: 3 4 5 6 7 8 9 10 11
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13 14
Number of prime knots: 1 1 2 3 7 21 49 165 552 2176 9988 ?
OBS: in this list Thistlethwaite doesn’t count both a knot and its mirror image. A knot that is
equivalent to its mirror image is an ‘amphicheiral knot’. If the knot is equal to its mirror image, no
information is lost in his list, but if the mirror image of the knot isn’t the same as the knot we have to
add this knot to his list.
(F.x. the figure eight-knot is amphicheiral, because if you, by using Reidemeistermoves, change every
crossing, to the opposite you get its mirror image. These two are the same knot and count as one knot
in Thistlethwaites list.)
This is the second main unsolved problem in knottheory: Prime knot enumeration
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Thilde Kamstrup-Jespersen
AkMat Blok 1, efterår 2008
We know how many knots there are up to 13 crossings, but not how many from 14 and forth. We also
know that the number of prime knots grows at an exponential rate and recently it was proved that the
number of distinct prime n-crossing knots is bounded above by an exponential in n.
In 1987 some theorists tried to prove that the number of distinct prime knots of n crossings is at least
(2n-2 -1)/3 for n ≥ 4.
Here both a given knot and its mirror image are counted if they are not equivalent. Therefore it’s
obvious that this number can exceed the numbers in Thistlethwaite’s list because some of the knots in
his list are not amphicheiral knots.
Knot invariants
There can be a lot of projections of the same knot and therefore knot-theorists search an entity that can
distinguish between knots even though the form of the knot changes.
F.x. they search an entity so that no matter what form the unknot has, this entity would be the same.
I’ve chosen 2 invariants: crossingnumber and polynomials.
Crossing number
The crossingnumber is the least number of crossings that occur in any projection of the knot. The
notation is c(K).
Determining the crossing number is difficult. First you have to find a projection of the knot with n
crossings. Then you know that c(K) is n or smaller. If all of the knots with fewer crossings than n are
known in the table of knots, and K doesn’t appear on this list the conclusion is c(K) = n.
F.x. 73 has c(K)=7, because it has a projection with 7 crossings and it’s distinct from all the knots of
fewer than 7 crossings.
But this gives us a problem: What is I have a knot with 15 crossings? Nobody knows how all the knots
with 14 crossings looks like and therefore this method is useless for knots with c(K)= n bigger than 14.
It’s always possible to determine c(K) of alternating knots. Start by calling a projection reduced if
there are no easily removed crossings. It’s easy to look at an alternating knot to see if it’s reduced or
not. An alternating knot in a reduced alternating projection of n crossings has c(K) = n, because there
is no projection of such a knot with fewer crossings. So it’s easy to tell the crossingnumber of an
alternating knot, but it’s not easy with a nonalternating knot.
This is another unsolved problem: How do we determine c(K) of nonalternating knots?
And another problem is: What is c(K) for a composite knot? Is the crossing number of K#L always
equal to the sum of the crossing numbers of the factor knots - c(K#L) = c(K) + c (L)?
The conjecture holds when the composite knot K#L is an alternating knot, but when it’s nonalternating
we don’t know.
Polynomials:
The most successful and interesting way to tell knots apart is by associating a polynomial to each knot.
The idea is that 2 projections of the same knot will give the same polynomial. So if you have 2 pictures
of 2 knots, computed the polynomials and they turned out different you can tell that the knots are
distinct.
The first polynomial applied to knots was done in 1928 by a guy named Alexander. And knottheorists
used his definition many years. Then in 1984 Vaughan Jones defined the Jones polynomial. IT was
very wild and crazy and hard to compute, so therefore Kauffman discovered an easier way to find a
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Thilde Kamstrup-Jespersen
AkMat Blok 1, efterår 2008
polynomial to distinguish the knots, and that is now known as the Kauffman/bracket polynomial,
which is the best known invariant. It can distinguish the knots up to 10 crossings..
First we have to develop the bracket polynomial, where we obtain a polynomial of a single variable A
and thereafter we can determine the Kauffman/bracket polynomial.
The rules for computing the bracket polynomial:
<K> is a knot projection, not a knot!!!
Simple examples:
The bracket polynomial for the trefoil knot:
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Thilde Kamstrup-Jespersen
AkMat Blok 1, efterår 2008
But the bracket polynomial is not yet and invariant because <O> =1 and
-A3. So the bracket
polynomial only works on projections and if it was an invariant these results were supposed to be the
same.
Kauffman then defined following polynomial for an orientated knot:
X(K) = (-A-3)w(K) <K>
Here w(K) is the writhenumber of the orientated projection of a knot. You put on a pile a place in the
knot, follow the string and count the crossings including the signs.
Now we have an invariant, since:
Unfortunately none of the invariants today are able to distinguish between all knots yet which leads us
back to the main problem.
Perspective:
Why do we have to care about this? What’s it worth working in this field?
Biochemists have discovered knotting in DNA molecules.
Synthetic chemists have realized it might be possible to create knotted molecules, where the type of
knot determined the properties of the molecule.
Significant applications to chemistry and biology and through that it contributes to the development in
these sciencefields which are very important to us today.
Knot theory is a big field and the development of the solutions to the unsolved questions can be very
useful and important in many ways and for many other sciences.
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