some colouring of knots

8/4/2019 Some Colouring of Knots

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Introduction:

Its hard to explain anything in knot theory without mentioning topology, after all, knot theory is part of

topology and they both have a lot in common:

Topology is a branch of mathematics that studies properties of objects that the object maintains

after a continuous deformation or morphing of it. As in knot theory, you can ask yourself whether two

shapes are different, or actually two representation of the same shape.

The objects that are of interest in topology are curves or surfaces in different manifolds (A

manifold can view as a surface with different amount of dimensions, i.e. Euclidian space would be a

manifold of three dimensions). These surfaces may in turn be embedded in a higher manifold.

A knot constitutes a smooth manifold (smooth means that the manifold is a differentiable manifold for

which all the transition maps are smooth), for instance, this would mean that the un-knot can be

perceived like a circle in R.

In topology two shapes that are the same are defined to be homeomorphic. Homeomorphism is

an isomorphism that portrays a constant topological invariant between two shapes.

A more formal definition of homeomorphism from Wikipedias page on the subject:

a homeomorphism or topological isomorphism or bicontinuous function (from theGreekwords

(homoios) = similar and(morph) = shape, form) is acontinuous functionbetween two

topological spacesthat has a continuousinverse function. Homeomorphisms are theisomorphismsin the

category of topological spaces that is, they are themappingswhich preserve all thetopological

propertiesof a given space. Two spaces with a homeomorphism between them are calledhomeomorphic, and from a topologicalviewpoint they are the same.

1

1 http://en.wikipedia.org/wiki/Homeomorphism
http://en.wikipedia.org/wiki/Greek_languagehttp://en.wikipedia.org/wiki/Greek_languagehttp://en.wikipedia.org/wiki/Greek_languagehttp://en.wiktionary.org/wiki/%E1%BD%85%CE%BC%CE%BF%CE%B9%CE%BF%CF%82http://en.wiktionary.org/wiki/%E1%BD%85%CE%BC%CE%BF%CE%B9%CE%BF%CF%82http://en.wiktionary.org/wiki/%E1%BD%85%CE%BC%CE%BF%CE%B9%CE%BF%CF%82http://en.wiktionary.org/wiki/%CE%BC%CE%BF%CF%81%CF%86%CE%AEhttp://en.wiktionary.org/wiki/%CE%BC%CE%BF%CF%81%CF%86%CE%AEhttp://en.wiktionary.org/wiki/%CE%BC%CE%BF%CF%81%CF%86%CE%AEhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Topological_spacehttp://en.wikipedia.org/wiki/Topological_spacehttp://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Isomorphismhttp://en.wikipedia.org/wiki/Isomorphismhttp://en.wikipedia.org/wiki/Isomorphismhttp://en.wikipedia.org/wiki/Category_of_topological_spaceshttp://en.wikipedia.org/wiki/Category_of_topological_spaceshttp://en.wikipedia.org/wiki/Map_%28mathematics%29http://en.wikipedia.org/wiki/Map_%28mathematics%29http://en.wikipedia.org/wiki/Map_%28mathematics%29http://en.wikipedia.org/wiki/Topological_propertyhttp://en.wikipedia.org/wiki/Topological_propertyhttp://en.wikipedia.org/wiki/Topological_propertyhttp://en.wikipedia.org/wiki/Topological_propertyhttp://en.wikipedia.org/wiki/Topological_propertyhttp://en.wikipedia.org/wiki/Topological_propertyhttp://en.wikipedia.org/wiki/Map_%28mathematics%29http://en.wikipedia.org/wiki/Category_of_topological_spaceshttp://en.wikipedia.org/wiki/Isomorphismhttp://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Topological_spacehttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wiktionary.org/wiki/%CE%BC%CE%BF%CF%81%CF%86%CE%AEhttp://en.wiktionary.org/wiki/%E1%BD%85%CE%BC%CE%BF%CE%B9%CE%BF%CF%82http://en.wikipedia.org/wiki/Greek_language


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The topological invariants are defined under continuous deformation of the shapes, this definition will

carry over to our knots, and well see that invariants in knot theory also are defined under continuous

deformation.

Tutorial:

The intentions of this section are to give a quick overview of the Alexander and the Jones polynomial.

We will start by talking about the need for a knot invariant, and give a quick introduction to the knot

polynomials. We will then move on to the Alexander and Jones polynomial in this order. The start of

each section for the polynomials includes a brief explanation of them, and goes on to describe rules and

methods for calculating the polynomial.

After the Alexander polynomial the Conway polynomial will be introduced, and an example will be

worked trough, so that the reader will be familiar with the skein relationship.

Even though we can calculate the Jones polynomial using skein relations, we will instead focus

on using the bracket polynomial; so there will be a tutorial on how to work with the Bracket and theKauffman polynomial, which in turn will be used to derive the Jones polynomial.

What makes a Knot a Knot, and what doesnt make it this knot?

Its quite easy to see that a trefoil and the un-knot are two different knots: You should be able to see

that its impossible to unravel the trefoil to form the un-knot (at least in a mathematical sense).

Before we continue to look at this we need some sort of a definition of what makes two knots the same:

Paraphrasing from the book knots and surfaces2; We define the concept of isotopy, or an equivalence

relation between knot diagrams: Two link diagrams (a projection of the knots on a plane in R3) are

isotopic if any one of them can be transformed into the other one by some succession of Reidemeister

moves R1, R2 and R3, and we call the two regularly isotopic if the transformation can be achieved

without using move R1.

2Knots and surfaces, N.D. Gilbert and T. Portes, Oxford university press (1995), p. 6


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The trefoil has a crossing number of 3

To calculate a knots Alexander polynomial you start with an oriented link diagram (i.e. a

projection where both orientation and crossings are marked out). A line connecting two crossings isknows as an arc. All crossings and arcs are then colored (the coloring of the arcs and crossing are

independent of one another). This coloring is then used to form an NxN matrix, where N is the number

of crossings. There are other ways to define the dimension of this matrix, for example, you could use the

number of areas that are enclosed by arcs instead of the arcs themselves. In turn, An Alexander matrix is

derived from this matrix, and the polynomial that forms the determinant is the Alexander polynomial for

that knot. It should be noted that an Alexander polynomial is only guaranteed to exist for a tame knot (a

knot that has a polygonal representative).

However, the Alexander polynomial is not a perfect invariant, as the resulting polynomial is very

much dependent on which link diagram of the knot is being used, or which method in the cases where

more than one is applicable. The polynomial for a given knot will differ with a factor of .4

A knot matrix can be determined by the following rules:

If a crossing 1 has a right handed orientation and got an arc A passing over the arcs B and C,then the element 1-t is inserted at position (1,A), -1 at position (1,B) and t at position (1,C) in

the NxN matrix.

If on the other hand the same crossing would have a left handed orientation, then the element1-t would be inserted at the same position. But the other two would be reversed

When the Matrix has been completed, you should delete the last column.4

Knots and surfaces, N.D. Gilbert and T. Portes, Oxford university press (1995), p.41


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The Remaining N-1xN-1 matrix is the Alexander matrix, and its determinant is the Alexanderpolynomial.

A right handed crossing and a left handed one respectively

Or we could by comparison use the region created between the arcs instead of the arcs themselves.

There is a relationship between the number of arcs and the number of regions that span the diagram,.

This relationship is defined by:

Thus, to calculate the Matrix for this invariant we need the following rules:

In a crossing with a right handed orientation these values get assigned to each of the followingregions surrounding the crossing

In a crossing with a left handed orientation these values get assigned to each of the followingregions surrounding the crossing

In case the crossing only has 3 distinct regions, i.e. a region spans more than one side of thecrossing; both values of those sides of the crossing should be assigned to that region in the

matrix.

When all the values have been inserted into the NxM matrix (where M = N+2), you shoulddelete two columns (regions) that are bounded by the same arc.

The Remaining NxN matrix is the Alexander matrix, and its determinant is the Alexanderpolynomial.


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Example:

We are going to attempt to find an Alexander polynomial for the figure 8 knot. We are also going to

attempt to show that this actually is an invariant, as it should yield the same result no matter what

method we use.

We will start by using the first method described above to solve for the Alexander polynomial of this

figure 8 knot:

To start with we identify the Arcs and the crossings; in this case the arcs have been colored by using

letters and the crossings by using numbers, the arrow represents the orientation. We now need to

assemble our matrix:

Step 1

-Examining the crossings:

Step by step we are going to move trough each crossing and see what arcs are part of the underpass and

overpass, and what orientation the crossing has. We are then going to insert these values into an NxN

matrix where the rows are the crossing number and the column are the arcs.

1st

The crossing is left handed and the values for the 3 arcs will be:

A = 1-t B=t C=-1 D=0


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2nd

The crossing is right handed and the values for the 3 arcs will be:

A = -1 B=t C=0 D=1-t

3rd

The crossing is right handed and the values for the 3 arcs will be:

A = -0 B=1-t C=-1 D=t

4th

The crossing is left handed and the values for the 3 arcs will be:

A = -1 B=0 C=1-t D=t


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Thus, if we collect all values for each crossing, we end up with the matrix:

M(A0)=

A B C D

1

2

3

4

Part 2

-Calculating the Alexander matrix.

This procedure is really simple, and is done by removing the last row and column from the matrix, giving

us a new N-1xN-1 matrix:

M(A1) =

Taking the determinant of this matrix now yields or Alexander polynomial:

Det(M(A1)) =

Now, instead of coloring arcs, we will attempt to color the regions that are surrounded by arcs and see if

it yields the same polynomial for the figure 8 knot:


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The orientation of the knot has been reversed in order to show that orientation doesnt affect the

invariant. The crossings are still colored with numbers and the regions are colored by roman numerals.

Lets start by calculating the Matrix by looking at the crossings (note that the crossings have the same

numbers as before, so were moving backwards starting at crossing 4):

Part1:

4th

This crossing is left handed, and the values retained for each of the following regions are:

I=-1 II=-t III=1 IV=t V=0 VI=0

3rd

This crossing is right handed, and the values retained for each of the following regions are:


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I=-1 II=0 III=t IV=0 V=1 VI=-t

2nd

This crossing is right handed, and the values retained for each of the following regions are:

I=--t II=0 III=0 IV=t V=1 VI=-1

1st

This crossing is left handed, and the values retained for each of the following regions are:

I=0 II=-t III=t IV=1 V=0 VI=-1

Adding these values together gives the matrix:

M(A0)

I II

III

IV

V VI

1

2

3


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4

Part 2

Now, in order to get the Alexander matrix, we need to remove any two columns that share the

same boundary arc, for instance, we could remove column II and IV to make it easier tocalculate the determinant of the matrix.

The matrix we end up with is:

M(A1) =

The determinant of the Alexander matrix is:

Det(M(A1)) =

Which is exactly the same determinant we got for the other method? But if we would have deleted row

I and V instead, and ended up with the Matrix:

M(A2)=

This has the determinant:

Det(M(A2)) =

Which is the same answer as the other, but with an extra factor of

It gets increasingly complicated to calculate the Alexander polynomial for knots and link diagram as the

number of crossings increases


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AlexanderConway polynomial

Discovered by J. Conway in the late 1960s, the Conway polynomial is defined trough a skein relation5 -a

relation between the polynomials of two different links, and can be used as an alternative way to

calculate the Alexander polynomial. It is not as direct as the matrix method for solving polynomials, but

it has the advantage of being a recursive function, so work on one knot may be used in another.

An Alexander polynomial of tame knot can be defined by the three following rules

trivial link = 0

This will be used depending on the alignment of the crossing that is being manipulated.

Where the last one is the skein relation; three projections of the links L+, L- and L0 are identicalexcept for in the regions below:

From this we can deduct the two equations:

With these rules we can try to calculate an Alexander polynomial by recursively choosing

crossings and using the skein relationship to try and obtain two simpler links. This will in turn reduce the

projection into a trivial projection. This process will produce whats known as a resolving tree, which

gives a very easy overview of what is happening.

Examplethe figure 8 knot:

We denote the knot by using the notation: We start by looking at the original knot; choose a crossing and identify its orientation:

5http://en.wikipedia.org/wiki/Alexander_polynomial


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The marked crossing has a right handed orientation. By using the skein relations we can break it down to

equation a from the third rule, giving us two new diagrams which we will label 1 and 2:

Diagram 1:

Applying the L- move we end up with:

This is a trivial knot (the un-knot) and reduces to a value of 1 by the first rule of the Conway polynomial.

Diagram 2:

Applying the L0 move we end up with:

This is a link diagram with two intertwined knots. We need to pick a new crossing and do further work to

reduced this to a trivial knot/link, the next crossing to be investigated has been marked on the previous

diagram:

Crossing 2:


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This crossing has a left handed orientation, and will be reduced by using equation b from the third rule.

This will in turn yield 2 new diagrams that we will label 3 and 4:

Diagram 3:

By applying the L+ operation we end up with two un-knots, or a trivial link diagram which value is zero in

the Conway polynomial.

Diagram 4:

The last operation L0 will morph the link diagram back to a trivial knot; we have finally managed to

reduce the figure 8 knot into a trivial knot.


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Finally we take all the operations that have been used for the reductions to produce the Conway

polynomial:

Plug in the values the reductions have yielded:

This is the Conway polynomial for the figure 8 knot.

Jones polynomial

A big brake trough in knot theory was made in 1984, when New Zealander Vaughan Jones introduced

the Jones polynomial as a new invariant for knots. This in turn led to the discovery of a deeper seated

connection between knot theory and quantum theory, hyperbolic geometry, methods in both statistical

mechanics and quantum field theory.

It was during his work on von Neumann algebraic6

(algebra of bound operators on Hilbert space)

he discovered polynomials that acted as invariants for knots. To begin with, he thought that what he had

discovered were Alexander polynomials. But on the contrary, it turned out that hed discovered whats

subsequently was generalized to the two-variableHOMFLY polynomials. Unlike the Alexander

polynomial, the Jones polynomial can assume different values for a knot and its mirror, for example:

The Jones polynomial for a trefoil is: While the Jones polynomial for its mirror is:

7

A trefoil knot and its mirror

6V. F. R. Jones,A new knot polynomial and von Neumann algebras, Bull. Amer. Math. Soc., 33 (1986)

7http://mathworld.wolfram.com/JonesPolynomial.html
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The simplest way to calculate the Jones polynomial for a knot is to use the same linear skein theory as in

the Conway polynomial. But instead of using the skein relation for the Alexander polynomial we get:

But instead of this we are going to look at a way to get the Jones polynomial from the Kauffmanpolynomial. These two polynomials are really the same, apart from a change of variables.

Before starting our work with the Jones polynomial we need to give a quick introduction to the

Bracket polynomial. This polynomial must be calculated in order to be able to get the Kauffman

polynomial, but is in itself not an invariant in R.

Definition:

The bracket polynomial of a diagram D is the Laurent polynomial in one variable A, defined by thefollowing rules:

Where mean that the relation can be applied to the part of the link diagram inside the

bracket.
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As mentioned earlier, in order to form an isotopy invariant the Reidemeister moves R1, R2 and R3 must

retain the same polynomial invariant independent of projection. The bracket polynomial forms an

invariant under regular isotopy (Riedemeister moves R2 and R3), but as will be shown now, it doesnt

form an isotopy invariant as it produces different results for move R1:89

There are several ways to bypass this problem:

We could find a number A so that . This would give a numeric invariant.The span of the bracket polynomial could be used, the span is the difference between the

highest and lowest power of A

We could use the writhe of a knot projection, the writhe of a link is the sum you get by addingup all the crossing of the link D; add +1 for a right handed crossing and -1 for a left handed.

We will use the notion of writhe for the Kauffman, and thus the Jones, polynomial.

To start with all we need to do is to change the projection D to D which is the same as D, but

it has an orientation. Then a simple renormalisation solves the problem and it is true that, given

K(D), where K is the Kauffman polynomial:

()

There is the bracket polynomial of the original un-oriented projection and is the writhe of theoriented projection.

8http://en.wikipedia.org/wiki/Bracket_polynomial

9N.D. Gilbert and T. Portes, Knots and surfaces, Oxford university press (1995), p. 22
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The Kauffman polynomial now forms an isotopy invariant. To get from the Kaufmann to the

Jones polynomial, all we need to do is to replace the As by , i.e: ()

Now we can try to find the bracket polynomial for the figure 8 knot:(The instruction have not yet been written)

We start by taking D, the un-oriented figure 8 diagram. We choose a crossing and break it up according

to the bracket rules, since we have a positive crossing, we get:

Now, its worth noting that the two diagrams brake down into two different shapes: the first one

appearing is the trefoil, and the second one becomes a trivial link after applying move R1 to untanglethe figure 8.

The reason we do this is that as we break down the trefoil, well run into a second trivial link, so instead

of spending time on solving this link twice, we can turn our focus to the trefoil and find the bracket

polynomial for it instead. You should be aware of the reusability of the bracket polynomial; if you solve

the bracket polynomial for one shape, it can be substituted for wherever it appears in other solutions

(do mind handedness, as most knots are chiral, i.e. there polynomial is different depending on their

handedness). It does require a bit of intuition to be able to spot where you can cut corners, it always

worth to try to quickly look at the different links produced and see if they are, or can be broken down

to, shapes that youve already have a bracket polynomial for.

The trefoil:

Again, we start out with the unoriented knot diagram for the trefoil, we chose a crossing and apply the

bracket rule. In this case we have another positive crossing, so we get:

The first shape can be reduced to a trivial knot by applying the R1 move twice, the second one is the

same trivial link that appeared in our original figure eight knot, so by solving it now, we can substitute
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the bracket polynomial back into our original equation. The crossing we are reducing on the un-link is

negative and will, together with the R1 move over the negative crossing on the first knot, yield:

This is the last step where we will have to bother with the diagrams; as we can see, the three remaining

ones can all be reduced to trivial knots with the R1 moves. The crossings are, in order from left to right:

Negative, negative , positive .This reduces to the following polynomial, where the bracketed circles are trivial knots:

By not multiplying the reduced trivial chain with, we get its bracket polynomial, which is:

Thus, all we need to do is to substitute the new found polynomial for the trivial link and the one for the

trefoil back into our original equation. This gives:

Now this bracket polynomial is not, as mentioned earlier, an invariant under isotopy, so in order to get

at the Kauffman polynomial we need to normalize with the writhe of the figure eight knot:
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The writhe is zero, plugging it into the equation for the Kauffman polynomial yields:

Our original bracket polynomial!

Finally, to get to the Jones polynomial we simply need to replace the variable A by :

This is the Jones polynomial for the figure 8 knot.

The Jones polynomial for the mirror of the figure 8 knot is:

This is identical to the previous polynomial! This means that the figure 8 knot is achiral, i.e. it lacks

handedness. Its left for the reader to calculate the Jones polynomial for the mirror of the figure 8 knot

to prove this.

References:

http://en.wikipedia.org/wiki/Alexander_polynomial

http://en.wikipedia.org/wiki/Topology

http://en.wikipedia.org/wiki/Jones_polynomial

http://en.wikipedia.org/wiki/Homeomorphism

Lecture notes from both compact surfaces and knot theory (http://www.warwick.ac.uk/~maaac/MA3F2-

page.html)

http://mathworld.wolfram.com/JonesPolynomial.html

V. F. R. Jones,A new knot polynomial and von Neumann algebras, Bull. Amer. Math. Soc., 33 (1986)

V.F.R Jones, The Jones Polynomial, Dep. of Math. Uni. Of California (2005)

N.D. Gilbert and T. Portes, Knots and surfaces, Oxford university press (1995)
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some colouring of knots

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