Knot Theory

• Intro & Terminology
• Isotopy, Reidemeister Moves & Invariants
• Homfly

Introduction

Knot theory incorporates many aspects of mathematics including geometry, algebra, topology, set theory and number theory. The study of knot theory has lead to advances in molecular DNA biology, quantum physics and statistical mathematics. The mathematical definition of a knot is given below which states that a knot is actually ‘knotted’.

Definition of a Knot:
A subspace K ⊂ ℝ³ is a knot if it is the image of a smooth injective map f : S¹ → ℝ³, with never zero.

When studying knots it's very difficult to visualise them in ℝ³ this is why we use a diagram of the projection of the knot on a smooth plane. This is explained in the above definition.

Examples of knot diagrams:

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You may find it useful to try to replicate these knots using pieces of string to observe their characteristics, of which will be explained below. Please note the Unknot is the only diagram which is not ‘knotted’ and looks like below:

Please note that when drawing a knot diagram there are certain rules which must be followed to ensure the knot diagram is not confused. For example the section of a knot diagram to the left would not be allowed as the arcs touch but not cross and it is not very clear. Also you cannot draw two crossing on top of each other. Diagrams should be clear and show crossings as they appear. You can imagine when drawing a knot diagram you are taking a shadow of the knot in a plane in ℝ³ so if the diagram is not clear you can take a slightly different angle of that plane where it is clearer.

Terminology

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Components- Number of separate classes in a diagram. i.e. the different pieces of string in your model. For example the Hopf link has 2 components but the trefoil only has one. Knots and links go hand in hand as a knot is actually a special link with only one component!

Arc - The continuous segment in a diagram.

Crossing – Where arcs meet. Every diagram will have a crossing number which is the total number of crossings. For example the trefoil has 3 crossings.

An Oriented Diagram – This is a knot diagram which has direction arrows on the arcs so that it is oriented.

At each crossing there is an overpass which is the unbroken arc at a crossing, and an underpass which is the broken arc at a crossing. Using this notation you can work out the orientation of the crossing, if the arcs have direction arrows. In general, when approaching a crossing along the underpass, if the overpass is travelling towards your left the crossing is positive. This can be seen more clearly below but is a very good thing to learn by heart.

When you have calculated the orientation of the crossings, you can then observe the ‘Writhe’ of the diagram. This is the sum of the signs of all diagram's crossings and is notated w(D).

Isotopy

These characteristics can distinguish between some knots for example the figure of eight has 4 crossings where as the Hopf link has 2 so you would believe that they are different knots but how can we prove this? Or how can we prove that two knots are the same? If two knots are the same we say they are isotopic, the formal definition of which is found below.

Definition of Isotopy:
Diagram D and D’ are isotopic if D can be obtained from D’ by a sequence of Reidemeister moves of type R0, R1, R2 and R3 or is Regularly Isotopic if R1 is not used.

Reidemeister Moves

These Reidemeister moves are as follows:

With experimentation you can see that these moves do not affect the knot's structure and hence preserve isotopy. The diagrams show just the local crossing area which is how the moves should be performed as they should not affect any other part of the diagram.

This definition only applies to the diagrams of knots so we need a way of linking it to the actual knots themselves. This is done by the following theorem hence diagrams are only isotopic if and only if their knots are isotopic.

Theorem:

1. Any link L is isotopic to a link which has a diagram D.
2. Given L₀ and L₁ have diagrams D₀ and D₁. Then L₀ and L₁ are isotopic if and only if D₀ and D₁ are isotopic.

Invariants

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Our overall aim in Knot theory is to classify knots into groups in which no two knots are the same, and hence form a complete list of knots. Therefore we need to be able to prove isotopy to make sure we get no duplicates in the list and this is done using invariants. Invariants are can take many forms but they are equal if and only if the knots are the same and it will not be altered by the Reidemeister moves! For example the writhe is an invariant but it cannot distinguish all knots from each other, this would be a complete invariant. This is why we need other invariant which when used together can distinguish more knots!

One of the main forms of invariants used is polynomials with the most common and popular polynomial being the bracket or Kauffman polynomial, but this polynomial is ony regularly isotopic. From this the Jones polynomial and Alexander polynomial were developed. But we are going to focus on the Homfly polynomial which although was developed last is the strongest polynomial in the fact that it can distinguish more knots than the other polynomials which act like special cases of the Homfly.

Homfly Polynomial

The Homfly polynomial gets its name from it's co-discovers: Hoste Ocneanu Millett Freyd Lickorish Yetter. It is sometimes referred to as the Homfly-PT polynomial which recognises the work contributed by Przytycki and Traczyk.

Definition of the Homfly Polynomial:
Let L be an oriented link. The Homfly polynomial for L P (L) (ℓ, m) is a unique two-variable Laurant Polynomial (Laurent polynomial allows positive and negative powers i.e. x⁻² + x⁻¹ + x + x²) defined by:

HP1) P (L) is an isotopy invariant.

HP2) P (U) =1 where U denotes the Unknot.

HP3) If L ₊, L ₋ and L₀ are three oriented links identical except at one crossing at which we see the configuration below then

ℓP(L₊) + ℓ ⁻¹P(L₋) + mP(L₀) = 0

Note HP1 allows the computation of P (L) from any diagram of L and HP2 and HP3 shows how to carry that computation out.

These computations (HP1, HP2, and HP3) produce a unique result and the proof of this can be found in Lickorish and Millett (1987).

We denote the c-component unlink by U^c. Since U¹ = U (the unknot), HP2 tells us P (U¹) =P (U) =1

Reference: Knot and Surfaces – Gilbert and Porter

Examples of the Homfly

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Let's consider U² – a 2 component unlink.

Using Reidemeister Move R1 we can see that L ₊ and L ₋ are unknots and so applying HP2, P (L ₋) =P (L ₊) =1

Therefore using HP3:

ℓP(L₊) + ℓ ⁻¹P(L₋) + mP(L₀) = 0

ℓP(U) + ℓ ⁻¹P(U) + mP(U²) = 0

ℓ(1) + ℓ ⁻¹(1) + mP(U²) = 0

ℓ + ℓ ⁻¹ + mP(U²) = 0

mP(U²) = −ℓ − ℓ ⁻¹

P(U²) = m⁻¹(−ℓ − ℓ ⁻¹)

P(U²) = −m⁻¹(ℓ + ℓ ⁻¹)

This is hence the Homfly polynomial for U². It can actually be shown that P (U ^c) = µ^{c −1}where µ = −m⁻¹(ℓ + ℓ ⁻¹). You may want to prove this yourself.

Let's now try a left hand Hopf Link.

(in this case we will operate on the top crossing which is negative)

From the diagrams you can see L₊ is equal to U² when an R2 Reidemeister move is performed. Therefore

P(L₊) = P(U²) = −m⁻¹(ℓ + ℓ ⁻¹).

You can also see that L₀ is actually the Unknot (when R1 is performed) hence P (L₀) =P (U) =1.

Consequently when we use HP3 this is what we get:

ℓP(L₊) + ℓ ⁻¹P(L₋) + mP(L₀) = 0

ℓP(U²) + ℓ⁻¹P(L₋) + mP(U) = 0

ℓ(−m⁻¹(ℓ + ℓ⁻¹)) + ℓ⁻¹P(L₋) + m(1) = 0

−ℓm⁻¹(ℓ + ℓ⁻¹) + ℓ⁻¹P(L₋) + m = 0

ℓ⁻¹P(L₋) = ℓm⁻¹(ℓ + ℓ⁻¹) − m

P(L₋) = ℓ²m⁻¹(ℓ + ℓ⁻¹) − ℓm

P(L₋) = l³m⁻¹ + lm⁻¹ − lm

Hence we have found the Homfly polynomial for the Hopf Link.

Note that to find the Hopf link polynomial we needed to already know the polynomial for U², you will find this a lot as you broaden the number of polynomials you complete. Therefore it is good practice to keep note of the polynomials you have completed and what they are so they you can use them again.

From the diagrams you can see that L + is actually the unknot when a series of R1 and R3 moves are preformed. This then implies that P (L₊) =P (U) =1. You can also see that L₀ is actually the Hopf link, and from before we saw the Homfly polynomial for this is ℓ³m⁻¹ + ℓm⁻¹ − ℓm.

Knowing this we can use HP3 to compute the Homfly polynomial.

ℓP(L₊) + ℓ⁻¹P(L₋) + mP(L₀) = 0

ℓ(1) + ℓ⁻¹P(L₋) + m(ℓ³m⁻¹ + ℓm⁻¹ − ℓm) = 0

ℓ + ℓ⁻¹P(L₋) + ℓ³ + ℓ − ℓm² = 0

ℓ⁻¹P(L₋) = −ℓ − ℓ³ − ℓ + ℓm²

ℓ⁻¹P(L₋) = −2ℓ − ℓ³ + ℓm²

P(L₋) = −2ℓ² − ℓ⁴ + ℓ²m²

P(L₋) = ℓ² m² − 2ℓ² − ℓ⁴

Mirroring the Homfly Polynomial –

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If the knot diagram is mirrored what happens to the Homfly polynomial?

Let L be an oriented link with mirror image L^!

Then P(L)(ℓ,m) = P(L^!)(ℓ⁻¹,m)

This is saying when the knot diagram is mirrored there is a mapping from each diagram of ℓ → ℓ⁻¹ when L → L^!.

Proof: Taking a mirror image reverses the signs of the crossings therefore the commutation of P (L^!) is the same as P (L) with ℓ and ℓ⁻¹ interchanged.

Why not try an example of this!