2 - y – 2 = 0 ⇒ y = 0
So x = 0, y = 0, z = 2, w = 1
But at c1, the relationship is 2z – y – x = 0, which becomes:
4-0-0=4
which is not 0 (mod 3), and so this labelling will not work.
What if we take w = 2, z = 1?
Then at c2, the relationship 2w – y – z = 0 becomes 4-y-1 = 0 ⇒ y = 0, but this makes the relationship 2z-y-x=0 at c1 become 2 – 0 – 0 = 2, which is not 0 (mod 3). Therefore, this labelling will not work either.
Since neither of these labellings work, we know that there is no labelling with
x = 0.
We continue in this manner, making x = 1, and then x = 2. It can be shown that neither value will provide a labelling for the knot.
Since the figure of eight knot cannot be labelled, mod 3, what can we do with it? How many numbers need be made available in order for a labelling to exist?
A similar method can be applied to the figure of eight, but instead of using mod 3, we can use a different modulo, in this case, mod 5. If we have a figure of eight, labelled as below, then there are 20 possible solutions:
Clearly, it is quite time-consuming to find all the labellings for a knot, and it may be possible that a knot, such as the figure of eight, can be labelled in another modulo. In order to compare the labellings of several knots, we decided to create a spreadsheet in which the calculations for each crossing of a knot would be worked out for us. (A guide to using the spreadsheet is at the end of this section). We could then use a test to decide if a labelling was valid or not. In all, we managed to test 14 knots (including the two above) for mod p labelling.
The following knots can be labelled as:
(Table shows which value of p we need to use to label the first 14 knots. The 76-knot showed no valid labellings for any value of p up to 51, with 50 000 labellings generated by the Excel spreadsheet for each value of p)
This table allows us to decide which knot we are dealing with, given that we know the number of crossings, and the value of p for the mod p labelling. For example, if we had a 6-crossing knot, and we manage to label it mod 3, then it must be the 61-knot. However, mod labelling is not foolproof. If we had a six crossing knot, and we manage to label it mod 11, then we could still not know whether it is the 62 or the 63 knot, as both these (distinct) knots can be labelled mod 11.
Further to this, we began noting down the labellings that were available for each knot. It may be that we could decide in advance how many distinct labellings there were for each knot. For each knot, we had to try many thousands of combinations, noting down the labellings that showed up as valid. For the first four knots, the values in the table are the only values available (see argument below).
The numbers in red are based on the following idea. Take the trefoil knot (31). The trefoil can be mod 3 labelled, and for each labelling, the arc called x can take one of three values, 0, 1, or 2. For each of these values of x, we can assign a value for arc y, however, y cannot have the same value as x unless all three values are the same (which is not allowed). Therefore, all the values are different. This means that, for each value of x, there are two labellings. This gives a total of six labellings for a knot that can be labelled mod 3. Note then, that 6 = 3 × 2 = modulus × (modulus – 1), or in other words,. This result, whilst not proven by the Excel spreadsheet, is borne out by repeated calculations of over 500000 labellings.
The same argument is applied to the figure of eight knot (41), but this time in mod 5. Here we can have the value for x as 0, 1, 2, 3, or 4. Each of these has four different labellings associated with it, giving a total of 20 labellings. Again, this is borne out (but not proven) by the Excel spreadsheet.
If this is the case, then, it is possible to calculate how many different labellings there are for a knot, given its modulus.
Using The Excel Spreadsheet
For Mod p Colouring
The Excel spreadsheet is designed in such a way that it can process thousands of randomly generated mod p labellings for knots in a short space of time, and that it can ‘flag up’ those labellings that are valid.
The following is a screenshot of the mod 3 labelling of a trefoil knot.
Cell A1 contains the modulus, p, we wish to use for the labelling. This can be changed to try different labellings, since a knot that cannot be labelled mod 3 may be labelled mod 5 (such as the figure-of-eight knot).
x, y, and z are the labels of the arcs, and c1, c2, and c3, are the labels of the crossings.
The formulae under the x, y, and z columns generate random numbers, which are rounded to integers less than the modulus, so in this case, the numbers generated will only be 0, 1, and 2.
The formulae in the cells below c1, c2, and c3, are the conditions that must be satisfied at each crossing, i.e., that , where a is the overpass, and b and c are the other two components of the crossing.
The formula in columns g, h, and i test that the conditions at each crossing are satisfied, i.e. that the value at each crossing is 0, or is divisible by the modulus. A ‘1’ appears if the conditions are met.
In column j, the values in column g, h, and i are added, and if the conditions are met at each crossing the number here should be the same as the modulus, p, (in this case, 3).
Column k is the column that shows the valid labellings. It works using a formula to check that the arc labels are not all the same (the numbers in columns a, b, and c), and that the number in column j is equal to p (in this case, 3). The formula is:
=IF(AND(NOT(AND(A3=B3,A3=C3,B3=C3)),J3=3),”valid”,””)
It is this formula that is the limiting factor to the number of crossings we can check, and hence the size of the knot we can work with on this spreadsheet. Excel only allows 30 items in an AND statement, and an 8-crossing knot would need 28 comparisons (possible), but a 9-crossing knot would require 36 comparisons which would be very difficult to calculate.
Given that not all three labels are the same, and that the condition is met at each crossing, the word “Valid” appears in column k. Using the filter button on column k allows the user to show only those labellings that are valid.
To recalculate the worksheet (without recalculating all the sheets in the workbook), press SHIFT+F9.
All the worksheets for the other knots work in the same way, but the formulae to test the conditions at each crossing will vary, depending on the knot.
Note that the worksheet for the 76-knot did not produce any valid labellings at all, even after many recalculations
Alexander Polynomials
There are many ways of describing a knot, such as the number of crossings, colourability, labelling etc., but all of these have a problem in that they may not describe the knot in such a way as to distinguish it from any other knot, or indeed, from the unknot. The search is on for the method that will do this, and J. Alexander took a major step in around 1928 when he invented the polynomial system, now known as Alexander Polynomials.
During the research for this project, we have come across three different methods of computing the Alexander Polynomial.
Method 1 – The Dot Method
(See: Casti (2000))
If we start on the first arc (labelled: start here), and follow the direction around the knot, every time we go under a crossing, we place two dots on the left hand side of our position, one as we go into the crossing, and one as we come out of the crossing. These dots will enable use to identify regions according to the following equation:
Using these criteria, we can build up a set of equations, one for each crossing.
At crossing 1:
rj = r0, rk = r2, rl = r4, and rm = r1.
Therefore, the equation is:.
At crossing 2:
rj = r0, rk = r3, rl = r4, and rm = r2.
Therefore, the equation is:.
At crossing 3:
rj = r0, rk = r1, rl = r4, and rm = r3.
Therefore, the equation is:.
For each crossing, then we get the equations:
Next, we create a matrix. The rows of the matrix will correspond to the crossings (so in this case, there will be three rows), and the columns correspond to the regions (so there will be five regions). The values in the matrix will be the coefficients of the various regions in each equation (The coefficient of r0 in crossing 1 will be x, the coefficient of r2 is –x, and so on).
For this knot, then, the matrix is:
If we take this matrix, and delete two columns, we have a square matrix. The determinant of this square matrix will form the polynomial. We can delete any two columns from this matrix if the columns represent regions in the diagram that have a dot in them at the same crossing.
In our diagram, there are dots in r3 and r0 at the crossing c2. Delete r3 and r0 from the matrix, and the new matrix (The Alexander Matrix) is:
and the determinant is:
The Alexander Polynomial of the Trefoil knot is
Method 2 – The Left Hand / Right Hand Method
(See: Livingston (1993), pg 48 - 51)
We need to look at each crossing in turn, and decide whether the crossing is left-handed or right-handed. If we let the arc that goes over the top of a crossing be arc i, and the arcs that go under the crossing are j and k (j enters the crossing, and k leaves it), then right and left-handed crossings are defined as:
Looking at crossing 1, we can see that it looks like this:
If we compare this to the two diagrams above, we can see that this is actually a right-handed crossing (rotate it 90° clockwise, and you’ll see), and i = 3, j = 1, and k = 2.
Looking at the other two crossings:
Crossing 2:
This is also a right-handed crossing, but with i = 1, j = 2, and k = 3.
Crossing 3:
Again, a right-handed crossing, but this time i = 2, j = 3, and k = 1.
Now, for each crossing, we have a value for i, j, and k. We now create a matrix, putting values into the columns dependant on their value of i, j, and k. The matrix will have as many rows as there are columns (in this example, there will be three rows).
If the crossing is right-handed:
If the crossing is left-handed:
Therefore, the matrix for this example will be built by putting:
So, the matrix is:
In this method, we reduce the matrix to a smaller square matrix by deleting the bottom row and the last column. The determinant of this new, smaller, matrix will give the Alexander Polynomial:
The smaller matrix is:
And so the determinant is:
The Alexander Polynomial of the Trefoil knot is
Method 3 – The Quadrant Method
(See: Gilbert & porter (1994), pg 38 - 41)
For this method, we are not concerned with whether a crossing is left- or right-handed, but as to whether we can align it with the following diagram:
Take crossing 1:
Now that the underpass is in the same direction as the diagram above, we can see that regions Q = t,
R = -t, S = -1, and T = 1.
Look at crossing 2:
Since the underpass is aligned with the diagram above, we know that P = -t, Q = t, R = -1, and T = 1.
Look at crossing 3:
This is already aligned with our orientation diagram, so we can see that P = -1, Q = t, S = -t, and T = 1.
We now have our values to put in a matrix. The columns of the matrix will be given by the regions P, Q, R, S, T, and the rows will be given by the crossings.
The matrix is then:
As before, we can reduce the matrix to make a square matrix. We do this by eliminating two columns. The two columns must be regions within the knot that have a common boundary. In this case, then, we could delete columns P and T, R and T, or S and T, (where P is the first column, and T is the last column).
Let’s choose to delete P and T, the first and last columns. We then work out the determinant of the reduced matrix to get the polynomial.
The new matrix is:
So the determinant is:
This is a different result from the ones obtained from the other two methods. However, Alexander has this problem covered. The results (which are usually different due to labelling differences) should only differ by a given factor of .
This is the case here. If we factorise the polynomial by taking out , we get:
We can conclude, then, that
The Alexander Polynomial of the Trefoil knot is
Any of these methods can be used to determine the polynomial of any knot. In most cases it should be possible to determine if the knot is a knot at all, rather than just a jumbled representation of the unknot.
If we use the dot method on this “knot”, we get the following equations:
In this case, some of the variables (r0, r1, …) are repeated with different coefficients. If we collect the coefficients together when we put them in the matrix:
If we choose to reduce this to a 4 × 4 matrix by deleting the second and third columns (representing r1 and r2), then the determinant is:
Dividing by a factor of (i.e. –1), we get the polynomial as 1, which is the polynomial of the unknot. Our “knot” was not a knot at all.
Celtic Knots
Many cultures have used knots in their art, or as religious designs. Some of these are incredibly intricate. It may be, however, that when designing the artwork, or religious symbol, the artist has used an unknot, or a very simple knot, and by way of twisting and turning the design made a complicated pattern. It is possible to untwist many of these designs using Reidermeister moves, and thereby making it much simpler to find the polynomial of the knot.
One example of Celtic art is the design below:
It is possible to simplify this slightly by using Reidermeister move number 1, which can be used to ‘twist out’ a crossing. In this case, we can use the same move four times, thus cancelling out 4 crossings:
What, then, is the polynomial for this knot? It is not possible to untwist this anymore, so we need to label the diagram, and calculate the Alexander Polynomial.
The following diagram has been labelled to use the dot method of calculating the Alexander Polynomial. The crossings are labelled in red, the regions in blue.
The matrix associated with these equations is:
Deleting the first and second columns, (representing regions r0 and r1) we get the following:
Clearly, this is a major problem when it comes to working out the determinant. However, using the software, Derive, the determinant of this matrix was calculated to be:
There must be a simpler way to calculate the polynomial of such knots. In real life, it is unlikely that a knot will be a simple knot. It is more likely to be a very tangled knot with many crossings. Even with software such as Derive, the logistics of calculating the polynomials would be too much to carry through.
The answer lies in two parts. The first is to simplify the knot as much as possible using the Reidermeister moves. The second is to split the knot into several smaller sections, and work on those individually. What is the connection between the polynomial of a simple knot such as the trefoil, and the polynomial of the knot formed when two simple knots (trefoils) are joined together?
The Polynomials Of Joined Knots
Suppose we have two trefoil knots that we wish to join. We want to know the polynomial of the new knot that is formed.
If we take the two trefoils and cut one of the arcs in each trefoil, and join the ends together:
We now have a new knot. Straightening it out a bit, it looks like this:
Using one of the methods shown previously it can be shown that the matrix for this knot is:
The determinant, and hence the polynomial of the knot is:
At this point, it is customary to factor out to get the polynomial in its lowest terms. In this case, factorising out –t gives the polynomial as:
Therefore, this is the polynomial of the new knot, the sum of two trefoils. However, the polynomial is not formed by the addition of the trefoil polynomials since in this case we have a polynomial of order 4, whereas the addition of order 2 polynomials will give another polynomial of order 2. The only way to get order 4 from order 2 is by multiplication. Is the new polynomial created by the multiplication of the polynomials of the smaller knots?
In this case it is.
Problem: The polynomial for the knot formed by adding two trefoils is also the polynomial for the 820-knot, which as its name suggests, has 8 crossings. (This problem also shows up later in the section on DNA knotting). However, we used all three methods for calculating the Alexander Polynomial for this knot, and came up with the same answer each time. It may be that a newer polynomial (such as the Jones Polynomial, or the HOMFLY polynomial) may be able to sort out this problem.
Is this true for the addition of other knots? If we look back to the Celtic design from page 26:
Clearly, the design can be split into four simpler knots; in fact, it is made up from four trefoil knots. Each trefoil will have the polynomial , and the completed knot we know has the polynomial
The polynomial of the large knot is of order 8, and that of the trefoil is of order 2. Since we are combining four knots, the multiplication of four order-2 polynomials should give us an order-8 polynomial:
Therefore, if we add two or more knots we must multiply their polynomials.
Is this always the case? If we try to join two knots in a different way, will we always get a new knot with a polynomial that is the product of the original polynomials?
In this case, the theory still holds, and the polynomial is:
Knots in DNA
It may be surprising to know that knots can be found in the strangest of places. Even in the very building blocks of human life, DNA.
DNA is made up of base pairs called Adenine, Thymine, Guanine and Cytosine. Adenine only pairs with Thymine, and Guanine only pairs with Cytosine.
DNA is formed by many of these base pairs attaching themselves to a Sugar-Phosphate backbone. Like in the diagram below.
What happens then is that each base finds its matching base pair. The two bases form a bond, making a base pair, with two sugar-phosphate backbones, like in the diagram below.
This is commonly called the DNA "ladder".
The ladder then twists (right-handedly) to form what is known as the Double Helix, as in the picture to the right.
Millions of these double helices intertwine with each other millions of times to form a cellular nucleus.
The strands of DNA intertwine and coil so many of them can fit inside of the nucleus. Because of this, knots are very frequently formed.
The body's cells have to replicate themselves because they die very quickly (especially red blood cells that live for about a day). The way cells replicate is by undoing the base pairs in the DNA strands, and making new matching base pairs. To do this it must ‘untie’ all those knots it has made by intertwining with each other. The cell manages this by using an enzyme called Topoisomerase (notice the first four letters, topo, as in topology).
The way in which Topoisomerase works is fascinating because it acts, very nearly, like the Reidermeister moves.
Here is an example of what Topoisomerase does:
If we take a basic trefoil knot, the Topoisomerase will change the sign of a crossing. This just means that where a crossing went over, it would now go under.
This is achieved by the enzyme cutting the knot at a crossing, and rejoining it either underneath or over the top, depending on the nature of the crossing beforehand.
We can now use a series of Reidermeister moves on the last diagram, and you will see that it is in fact the unknot.
Obviously, the ‘separation’ technique that Topoisomerase uses is not allowed in knot theory, because you cannot break an arc, but the ideas are similar.
In more complex knots with more crossings, you will probably have to make more cuts to reach the unknot. The number of cuts it takes is called the unknotting number. This number can be used as a very basic method to describe the complexity of a knot (i.e. the higher the unknotting number, the more complex the knot).
In our trefoil example, the unknotting number is 1, because it only takes one cut.
Unlike colourability, polynomial labelling or mod labelling, it is nigh on impossible to compute the unknotting number without having to do it for each specific knot. For example, if you tried to compute the unknotting number for a knot with 6 crossings, your answer may well be wrong. This is because not all 6 (or whatever number) crossing knots will have the same unknotting number. Here are two that prove this.
None of this may seem of much importance, but in fact, it is used all the time in scientific laboratories. This is because the unknotting number tells us the number of times that the Topoisomerase has to act on the knot. In turn, this helps scientists determine rates of reactions for Topoisomerase.
In chemical production, if the scientists know that the product they are trying to make is a very complicated one which has many knots to undo, then they know that it is going to take a while for the Topoisomerase to complete its job. Therefore, knot theory has an important role to play in the world of science, as well as the mathematical world.
Finding the Alexander Polynomial for a DNA strand.
We will find the polynomials for a strand of DNA using the Alexander method. We wondered if it would be possible that one day, science will be able to find a polynomial for a complete cell, like a hair cell or a skin cell, just by using the DNA strands.
Here are three examples of DNA knots. The first is the simplest knot, the trefoil, which has the Alexander Polynomial
The second knot is a little more complicated:
What is the Alexander Polynomial for this knot?
We will use the Quadrant Method for determining the Alexander Polynomial, and so we need to label each region of the knot with the letters P to W. (Note that there is a region at V, but the diagram does not show this clearly). We then give the knot an orientation, and label each crossing in turn.
The matrix derived from this labelling, is as follows:
Therefore, the Alexander Matrix, the determinant of which gives us the polynomial, is:.
The determinant is calculated using Derive:
Here we can factor out -t to get the polynomial in its lowest terms, and so we get:
This is the Alexander Polynomial for the 61-knot.
The third DNA knot:
The Alexander Polynomial for this knot can be created in two ways. We could use any of the three methods on the whole knot, in which case the polynomial is (which we can factor down to ), or we could look at the knot as the addition of two trefoils, in which case we multiply the polynomial of the trefoil by itself: .
There is a problem here, however (which we noted on page 30, The Polynomials Of Joined Knots). The polynomial is not the polynomial for a six-crossing knot. It is the polynomial for the 820-knot. Does this mean that the Alexander method cannot tell the difference between the reef knot (which this knot is), and the 820-knot? This is entirely possible, as the Alexander method cannot distinguish between left- and right-trefoils, for example. A stronger method of computing polynomials may be needed.
Kauffman's Construction Of The Jones Polynomial
Considering the problems that Alexander Polynomials have in distinguishing some kinds of knot, we decided to look at a different polynomial altogether. The Jones method, created by Vaughan Jones in 1984 manages to distinguish between the left- and right-trefoils, and may be able to distinguish between the reef knot and the 820-knot that showed up as a problem earlier.
The method is as follows:
A crossing in a knot is made up of the overpass, and the underpass. If we imagine that the overpass is rotated anticlockwise, over the underpass, two of the four regions around the crossing (directly opposite each other) will be reduced in size. We need to label these regions as A. The other two regions we label as B.
We continue this labelling for each crossing in the knot.
Consider the trefoil. By this rule, it can be labelled like this:
We now need to consider how we use these crossings. If we again look at a single crossing, it may look like this:
With each crossing, we can now make the following changes:
We could delete the crossing, and join the loose ends so that there is an open path between the regions marked A.
We could delete the crossing, and join the loose end so that there is an open path between the regions marked B.
In any knot, then, we can change each crossing in 2 ways. This means that, in a knot with n crossings, we can create 2n new knot diagrams.
In each of these new diagrams, some (or all) of the crossings will join the A’s together, and some (or all) will join the B’s. We can then draw new diagrams as in this example:
Since we have joined the A regions at each of the three crossings, in this example, we will call this new diagram AAA. Hence, a diagram that has two crossings that join the A’s, and one that joins the B’s will be denoted AAB (or ABA, or BAA, depending on which crossing the B’s are joined at).
Recall that for a knot with n crossings, we would have 2n new diagrams. Since we have a trefoil, with three crossings, we will have 23 = 8 new diagrams, depending on whether we join crossings A or B. The new diagrams for the trefoil are shown overleaf:
In each of these diagrams, we count the number of ‘circles’, that is, unbroken loops. The number of circles is the state, s, of the diagram.
If the number of type A crossings in a diagram is a, and the number of type B crossings is b, then we can now calculate the polynomial for the knot.
For each ‘state’, we complete the following formula:
The polynomial is the sum of all these state formulae:
For the trefoil, then, the polynomial is:
Clearly this can be made simpler by cancelling out the A’s, collecting like terms, etc:
The Left Hand Trefoil
The left-handed trefoil is different to the right-handed trefoil, but we have seen previously that the Alexander Polynomial for the two knots is the same, t2-t+1. Using Kauffman’s construction of the Jones Polynomial, however, we can tell the two apart. The difference in the two begins with the labelling:
The state diagram ABA for each of these is going to be very different:
The terms in the polynomial are similar, but ‘swapped around’ because of this difference, and the polynomial for the left-handed trefoil is:
which differs from the right-handed version as the indices are the negatives of those in the first polynomial.
The Reef Knot / 820-Knot Problem
We are fairly confident that Kauffman’s construction would resolve the problem of the reef knot and the 820-knot that we mentioned earlier. However, to use Kauffman’s construction it is required to ‘draw’ all the different states of a knot diagram. There are 2n different states for a knot with n crossings, which would mean that the reef knot would have 64 states, and the 820-knot would have 256. Clearly, this is prohibitive without the use of, perhaps, a computer program.
Tie Theory
After being introduced to the theory of tie knots, and Studying the book The 85 Ways To Tie A Tie, (Fink & Mao, 1999), we thought it would be of interest to link the tying of ties to the work we have already done. Fink & Mao explain quite well the links between the two, and also that some of the tie knots can be deformed into the trefoil, or the unknot, The free ends of the tie (even if joined) are made to pass back through the knot, and the knot can be unravelled. They then go on to describe the tie knots, using their own terminology to describe them, and to show they are distinct. They manage this by pegging the top loop (around the neck), and the bottom loop (near the waist) to stop the knots from being pulled apart.
However, we decided that we would allow the loops to slip through the knot (this ‘slipping’ would be a Reidermeister Move, R2), and investigate, using Reidermeister Moves, exactly what the tie knots are, the unknot, the trefoil, or something else. What’s more, it should be possible to create a list of the Reidermeister Moves needed to unravel the knots into the simplest form possible. This list may show how many Reidermeister Moves are needed to reduce the knot to its lowest form, in effect, how ‘far away’ the knot is from the simplest form.
For example, the first knot in Fink & Mao’s book is the Oriental:
The Oriental Tie Knot ➔ 2×Reidermeister Move 2 ➔ Reidermeister Move 1 ➔ The Unknot
Therefore, the Oriental is only 3 Reidermeister Moves away from the unknot (if we include R0 to make it a perfect circle), and its Alexander Polynomial is 1.
The second knot in the book 85 Ways To Tie A Tie, is the Four-In-Hand, as follows:
The following table shows the first 10 tie knots in the book (11 if you count the Diagonal as distinct from the Kelvin). The second column shows the Reidermeister Moves needed to reduce the knots to their simplest form. The third column shows what conventional knot the tie knot becomes, and the fourth column shows the Alexander Polynomial for the knot.
Summary
We have learnt a great deal about knots through this assignment. We had briefly met the idea that knots could be 3-coloured, but we don't think that the ideas of knots having such complex structures, polynomials and uses had been fully comprehended before. We certainly had not realised that knot theory was such a huge topic, and that there was so much material around about the topic.
We particularly found knots in DNA and the battle to understand the various polynomials interesting.
In the DNA section, we found the idea of the Topoisomerase acting like Reidermeister moves very surprising, and the fact that every cell in the human body will contain a knot, because the DNA knots itself.
One thing that has been of great use is that we now know how to tie a tie 85 different ways, and we can also tell you whether the knot in your tie is indeed a knot or not.
A major finding in this assignment is the vulnerability of the ideas and concepts of knot theory, specifically in polynomials of knots. Whilst constructing polynomials for knots, we came across the knot that is the addition of two trefoils. The polynomial given was that for an 8-crossings knot, the 820, but there are only 6 crossings in the 2-trefoil knot. This uncovers a flaw in the Alexander polynomials, and similar problems may occur with the other polynomials too.
This means that there is still some work to be done with knot theory, and there may be many more knotty problems to be unravelled!
End Notes
In order to find these labellings we took the simplest knot, the trefoil, and by hand, worked through each possible labelling. For example, If we labelled one arc as 0, then the other arcs must be 1 and 2, etc. This was quite a lengthy process, especially when we tried to label the Figure-of-eight knot (or show how the Figure-of-eight knot cannot be labelled mod 3). Because of the time factor, we decided to ‘automate’ the process with the Excel spreadsheet. The building of the spreadsheet took some time, but once complete, we could check thousands of combinations very quickly.
(it is not important which of these labels go where, however, for the sake of convention, we have labelled the first arc on the right of the knot as x, the arc immediately following this, moving clockwise around the knot, as y, and so on. The first underpass you reach as you follow arc x, is labelled as c1. The next underpass is c2, etc. This convention is followed for all the knots in this section).
These solutions were found using an Excel spreadsheet, designed by us to calculate mod labellings. The working of the spreadsheet is described at the end of this section, but suffice to say that we haven’t proved that these are the only solutions, however, the spreadsheet tries 1000 random labellings per calculation, and it is possible to recalculate many times in a short period of time. These are the only solutions in, perhaps, 30 000 calculations.
The formula took several forms before we found the one that worked. We needed to be sure of several thing in order for the labelling to be valid, and we needed to ring them all together in this one formula, hence the nested operations.
This was probably the most difficult part of this project. We had to decipher what the various texts meant when they were describing the methods. Different books use different terminology (and ultimately, different methods entirely!). The Alexander Polynomial does not stand alone, and a lot of background reading was required merely to understand what the method required.
The biggest problem was in the labelling of the arcs or regions. In some cases, the labelling was vital, in other cases it was irrelevant. Eventually we settled on a labelling of our own devising (described in the report) which appeared to work for all the methods. However, it wasn’t until we worked out how to use Derive to calculate the determinants of large matrices, that things became a little clearer, and we started to get sensible results.
This required a little research in order to find out how the determinant of a larger matrix is calculated. Working with 2×2 matrices was straightforward, but the method of expanding the matrix by a row, or a column, was not.
To align the crossing with the diagram, it is only necessary to align the direction of the underpass, not the direction of the overpass. In addition, we had problems when we labelled the regions in the knot incorrectly: we initially labelled the region of the knot that fell between the two ‘outgoing’ arcs at a crossing with the label between the two arrowheads on the diagram. Results from this incorrect method were completely wrong.
This method relies on taking two columns out of the matrix. These two columns correspond to regions that have a common boundary. What is never explained in the text is that these regions must be regions inside the knot, you can’t take out the outside region of the knot. Several attempts to make this method work fell apart because of this.
This was another useful piece of information that seemed to be buried amongst other things. Many of our initial attempts to work out Alexander Polynomials were in fact correct, but for this factor of ±tn. Making the connection (eventually) solved several of our problems with the methods.
This method of collecting the coefficients together is not mentioned at all in the Dot Method for calculating Alexander Polynomials. There is reference in Livingston (1993, p49) to collecting coefficients, but it was not immediately obvious that it would also apply to this method. We decided to try it here anyway, and the result is correct.
We had discussed what we thought might happen if two or more knots were added together long before we found references to it in the texts. We had thought that addition of knots might mean addition of polynomials. Once we had the Alexander Polynomial methods worked out, we calculated the polynomial of the joined knot. It was here that we noted that we must multiply the polynomials in order to achieve the correct degree.
It wasn’t until much later, when working on the knots of DNA, that we noticed that the knot we had had the polynomial of an 8-crossing knot, while the knot only had 6 crossings. We have since attempted to work out the Jones Polynomial of the reef knot (The one with 6 crossings), but it has so far proved a difficult task. There are 64 different states to consider, and even with a computer, we haven’t yet found a polynomial that makes sense. It would be ideal if we could find the correct polynomial (Jones) for this knot, and for the 820-knot to see whether Jones got the problem sorted out.
We cannot be sure that this is correct, none of the texts have lists of the polynomials of knots above 9 crossings.
Our initial understanding of this method had not used this idea of labelling. We had assumed that we could delete and reconstruct the crossings in any order. It took quite a while for us to associate the labelling with the method for forming the states.
This wasn’t easy to find in that it didn’t seem to be a related topic. Labelling of knots is reported in most books, but not in the same sections as the polynomials, and so the links are difficult to see.
There is a description of how the term is derived in The Knot Book (1994, pg 150), however, most other texts just give the term as an end result, one of the basic rules for the generation of a Jones Polynomial.
This was the first result we got for the Jones Polynomial. We knew it was not the same as the Jones Polynomial in the texts for the trefoil. It wasn’t until later that we realised that the trefoil, and the left-handed trefoil would have different Jones Polynomials.
Knot 8 is a version of the Half-Windsor, and so it is no surprising that they are both the unknot. Perhaps the 2 X-moves are really Reidermeister Move number 2?
