Introduction to theory

Summary of the lecture by Gregor Schaumann 2016

[email protected]

Fakultät für Mathematik, Universität Wien, Austria

This is a short summary of the lecture “Introduction to knot the- ory”, held by the author in the summer term of 2016 at the university of Vienna. The summary does not replace a script of the lecture, it is aimed at giving an overview on the topics that were covered during the course. It provides you with references and some suggestions for further reading.

1 Introduction

Knots are topological objects familiar from daily experience. Two natural ques- tions arise:

(i) What is a good mathematical concept of a knot? When are two knots considered to be equivalent?

(ii) Can one classify all possible knots? Can one distinguish knots?

It turns out, that classifying knots is very hard, focusses mainly on tools to distinguish knots, so called knot invariants. In modern physics, knots arise as worldlines of quantum particles in a 3 dimen- sional topological quantum field theory. Thus, such a theory in particular leads to knot invariants. A mathematical definition of these are the so called quantum invariants of knots. A famous work of Witten [Wit89] gives a physical reason why the should be considered as a quantum invariant. This was made mathematically precise in [RT91] and [Saw96].

Plan of the lecture We define knots, their diagrams and a reasonable notion of equivalence between knots. Then we describe some basic phenomena that we observe when applying the definition. Among those the notion of connected sum of knots. Considering first examples of knots we are lead to the use of knot tables which lists knots that are prime with respect to the operation of connected sum. Indeed, every knot has a unique prime decomposition in prime knots. This is proven using Seifert surfaces in the second part. In the third part we start with a systematic investigation of knot invariants. Here, the Reidemeister moves are very useful, since they describe precisely the equivalence relation on knots using only knot diagrams. With this result we consider several combinatorial invariants, the knot colourings and discuss their geometric meaning using the knot group. The Alexander polynomial is intro- duced and its relation to the knot colourings is discussed. Finally, an alternative definition of the Alexander polynomial using the Conway-Alexander skein rela- tion motivates the definition of the Jones polynomial via the Kauffman bracket. This is used to prove the Tait conjecture. We do not just want to list one interesting invariant after the other, but we seek relations between the invariants and want to see in what sense they fall into certain classes of invariants. As important class we define quantum invariants in three steps: First we develop a diagrammatic calculus, then define tangles and finally define quantum invariants as devices to apply the diagrammatic calculus to tangles. We thereby avoid the (proper) languague of categories and functors due

2 to time reasons, but instead use an explicit presentation of tangles via generators and relations. Finally, Vassiliev invariants provide another systematic treatment of knot in- variants. Its relation to quantum invariants constructed from Lie algebras is mentioned at the end of the course.

1 Phenomenology of knots

As reference for this section, see [CDM12, Chapter 1].

1.1 Knots, what are they? We first define knots using smooth manifolds and embeddings.

Definition 1.1. A parametrized knot is an embedding φ : S1 → R3, where S1 = {(x, y) ∈ R2|x2 + y2 = 1} is the circle.

We always the orientation of circle that is counter-clockwise. This gives also an orientation to every parametrized knot. A reparametrization φ0 of a parametrized knot φ is an orientation preserving diffeomorphism diffeomorphism f : S1 → S1 such that φ0 = φ ◦ f.

Definition 1.2. An oriented knot is an equivalence class of parametrized knots under reparametrizations.

The following operation captures the deformation of one knot into another.

Definition 1.3. Let φ1 and φ2 be two parametrized knots. An isotopy F : φ1 → 1 3 φ2 is a smooth map F : S × [0, 1] → R such that

1 (i) F (x, 0) = φ1(x) for all x ∈ S ,

1 (ii) F (x, 1) = φ2(x) for all x ∈ S , (iii) F (−, u): S1 → R3 is a parametrized knot for all u ∈ [0, 1].

Lemma 1.4. Isotopy is an equivalence relation.

Definition 1.5. The set of isotopy classes of oriented knots is denoted K. Its elements are called knots.

3 1.2 Knot diagrams Definition 1.6. A knot diagram for an oriented knot φ is a plane P ⊂ R3 with a projection π : R3 → P , such that π has at most finitely many double points on the knot. On each double point, the knot has to intersect transversally. The double points are over- and undercrossings. A knot diagram is called (i) alternating, when over- and undercrossings alternate when travelling across the knot diagram. (ii) reducibe, when it becomes disconnected when removing a small neighbour- hood of a single crossing.

1.3 Operations on knots

1 1 1 3 3 Denote by σS1 : S → S the orientation reversal of S and by τR3 : R → R the reflection along some plane in R3. This defines two operations on oriented knots φ : S1 → R3:

(i) orientation reversal φ 7→ φ ◦ σS1 ,

(ii) mirror φ 7→ τR3 ◦ φ. This is well defined on equivalence classes K ∈ K and defines operation K 7→ σ(K) = K∗ and τ(K) = K. σ and τ form an action of Z/2Z × Z/2Z on K, where the generators 1 × 0 and 0 × 1 act via σ and τ, respectively. (That this is an action means just that σ2 = id = τ 2 and σ ◦ τ = τ ◦ σ) Definition 1.7. A knot is called (i) totally symmetric if all of K, K∗, K and K∗ are different, (ii) invertible if K∗ = K, (iii) plus-chiral if K = K, (iv) minus-chiral if K = K∗, (v) fully symmetric, if K = K∗ = K = K∗.

Further operation: Connected sum of knots K1#K2 is defined by joining the two knots together. Knot tables list knots up to the operations σ, τ and connected sum (just prime knots). Definition 1.8. A knot is prime, if it is not the connected sum of two non-trivial knots.

4 2 Knots and surfaces

We discuss two interplays of knots and surfaces: First knots on the torus, then surfaces with a knot as boundary. See [Sul00] as introductory reference.

2.1 Torus knots Knots on a surface: Consider T 2 = S1 × S1 the torus. The (p,q)-torus knot for p, q two coprime integers is the embedding S1 → T 2 that winds p times around the meridian (first S1) of T 2 and q times around the longitude (second S1) of T 2. Lemma 2.1. Every embedding of S1 into T 2 is isotopic to a (p, q)-torus knot for some values of p and q.

2.2 Seifert surfaces See [Sul00] for reference. Definition 2.2. A for an oriented knot K is an embedded oriented surface F in R3 with the knot K as boundary, ∂F = K. The Seifert construction gives a recipe how to construct a Seifert surface for every knot.

Definition 2.3. The genus g(K) of a knot K ∈ K is the minimal genus g(FK ) among all Seifert surfaces FK for K. Recall, that the genus of a surface counts the number of holes of the surface (the sphere has genus 0, the torus genus 1, etc.). Theorem 2.4. (i) The genus detects the unknot: For a knot K, g(K) = 0 if and only if K is the unknot.

(ii) The genus is additive under connected sum: g(K1#K2) = g(K1) + g(K2). A consequence is: Theorem 2.5. Any knot can be written as the connected sum of prime knots. Further Reading 2.6. For more details on Seifert surfaces, also in higher di- mensions, see [Rol76]

3 Invariants of knots

Definition 3.1. Let S be a set. A ν with values in S is a function ν : K → S.

Main principle: If for an invariant ν, ν(K1) 6= ν(K2), then the knots K1 and K2 are not equivalent.

5 3.1 First examples

• The genus is an invariant g : K → N0,

• The crossing number c : K → N0 is the minimal number of crossings among all diagrams for a given knot.

• The unknotting number u : K → N0 is the minimal number of crossings that is required to change any diagram of a given knot into a diagram for the unknot.

c(K) The relation u(K) ≤ 2 holds.

3.2 Reidemeister moves Given two knot diagrams, how can we tell whether they represent equivalent knots? First, they could be just deformations of one of the other without changing any crossing. This is formalized by the notion of ambient isotopy: Two knot diagrams D and D0 are called ambient isotopic, if there exists a smooth map F : R2 × [0, 1] → R2 with F (−, 0) = id and F (−, 1)(D) = D0. The Reidemeister moves ΩI, ΩII and ΩIII (f.e. [CDM12, Thm 1.3.1]) are three moves on a knot diagram, that change only a small part of the knot diagram. However, by repeated use of these moves one can pass between all diagrams for a given knot and all diagrams for all equivalent knots. In between might

Theorem 3.2. Two knots are equivalent if and only if two of their diagrams are related by a finite sequence of Reidemeister moves and ambient isotopies. In this case, all of their diagrams are related this way

The proof uses the notion of piece-wise linear knots and ∆-moves between them. Consequence: If we have a function on oriented knots, that stays the same if we perform an ambient isotopy and any of the Reidemeister moves on a knot, than it defines a knot invariant. We next define combinatorial invariants that are easy to compute and allow to distinguish knots in a practical fashion.

3.3 Knot colourings The topics in this subsection can be found in [Liv93] First we pick a set of three colours Col3 = {Red, Blue, Y ellow}. We call an arc in a knot diagram a segment of the diagram from an undercrossing to an undercrossing.

6 Definition 3.3. A knot diagram DK for a knot K is called colourable, if each arc of DK can be coloured using the three colours of Col3, such that (i) at least two colours are used,

(ii) at any crossing at which two colours appear, all three colours appear. Theorem 3.4. Being colourable or not is a knot invariant. Col : K → {±}. If a knot K is colourable, the value of the invariant is Col(K) = +, otherwise Col(K) = −. This amounts to check that the being colourable is preserved under the Reide- meister moves. In the proof we see that once one side of a Reidemeister move is coloured, then there is exactly one possible choice to colour the other side of the move (the “external edges” keep the colour before and after the move). Hence we obtain Corollary 3.5. The number of possible colourings of a knot is an invariant #Col : K → N0. Another consequence Corollary 3.6. The unknot is not colourable (Col(unknot) = −). It follows, that any colourabel knot is nontrivial, i.e. not isotopic to the unknot.

3.4 n-colourings and the knot group In order to generalize the invariant of the last section, we first reformulate it: If we use the colouring set Col3{0, 1, 2}, the conditions in the definition of colourability become (i) at least two numbers are used,

(ii) at a crossing where the overcrossing arc is coloured with x, the two other arcs with y and z, the condition

2x − y − z ≡ 0 (mod 3) (3.1)

is satisfied. (Recall, that x ≡ q (mod p) means that there exists z ∈ Z such that x = z · p + q)

We immediately generalize this to p-colourability, with Colp = {0, 1, . . . , p − 1}:

Definition 3.7. A knot diagram DK for a knot K is called colourable mod p, or p-colourable for a prime p > 2, if each arc of DK can be labelled with elements in Colp, such that (i) at least two numbers are used,

7 (ii) at any crossing where the overcrossing arc is coloured with x, the two other arcs with y and z, the condition

2x − y − z ≡ 0 (mod p) (3.2)

is satisfied.

Theorem 3.8. Being colourable mod p or not is a knot invariant. Colp : K → {±}. If a knot K is colourable, the value of the invariant is Colp(K) = +, otherwise Colp(K) = −. Again one can show, that the number of possible p-colourings is a knot invari- ant.

Use of matrices for colourability Using matrices to investigate colourability motivates the Alexander polynomial, which can be seen as a generalization of colourability. To check whether a diagram DK is colourable mod p, we

(i) label each arc with a variable xi,

(ii) label the crossings with ci,

(iii) the relations 2xi − xj − xk ≡ 0 mod p on each crossing define an n × n- matrix Col(DK) with columns the variables xi and rows the cj. Here, n is the numbers of arcs which is the same as the number of crossings. In each row there is exactly one 2, and two 1’s as entries, the rest of the entries are zeros.

(iv) The diagram DK is colourable mod p if and only if the equation

Col(DK ) · x ≡ 0 mod p (3.3)

has a solution with xi ∈ {0, 1, . . . , p − 1} and at least two xi different. We will show in the next section, that

Lemma 3.9. (i) Any of the rows and columns of Col(DK ) is a linear combi- nation of all the others with non-zero coefficients. If we erase an arbitrary tunc row and column we obtain an (n − 1) × (n − 1)-matrix Col (DK ), the truncated colouring matrix.

trunc (ii) The knot K can be coloured mod p if and only if Col (DK ) · x ≡ trunc 0 mod p, which is the case if and only if det(Col (DK ) is divisible by p.

8 knot group We introduce the knot group as the “universal colouring group”. First we generalize p-colourability further. Definition 3.10. Let G be a group. A knot diagram can be labelled by G, if there is a labelling of each arc by an element x ∈ G, such that (i) the labels generate the group G,

k

h (ii) at each right handed crossing g , gkg−1 = h, at each left handed k

g crossing, h , ghg−1 = k.

We can turn this around and define a group G(DK ) for each knot diagram DK by assigning generators to the arcs of DK and the relations of Definition 3.10 ii) for all crossings.

Theorem 3.11. The group G(DK ) is isomorphic to the fundamental group of R3 \ K and hence independent of the diagram up to group isomorphism. A knot is colourable by a group H if and only if there exists a surjective group homorphism G(K) → H. If we have such a homorphism φ : G(K) → H, we label an arc x by its value φ(x) ∈ H (recall, that the arcs generate G(K)) to obtain the label in H. Here we use the following results from group theory (i) Definition of free group and its properties, (ii) Presentation of any group by generators and relations, characterization of group homomorphism out of a presentation. p 2 We find again p-colourability by using the dihedral group Dp =< r, s|r , s , srsr >.

Lemma 3.12. Let DK be a knot diagram that is coloured by Dp. Then all labels ai on the arcs of the diagram are of the form sr with ai ∈ {0, . . . , p − 1}.

 ai Proof. The elements of Dp can be brought into a normal form of the type s r with  ∈ {0, 1} and ai ∈ {0, . . . , p − 1}. It is easy to see, that the elements of the form srai are invariant under conjugation, i.e. x · srai · x−1 is again of this form. According to Definition 3.10, the labels on the undercrossing are conjugated by the label of the overcrossing. Hence, if the label on one arc is of the form srai , the labels on all arcs are of this form. But since the labels generate the group, there has to be a label with a factor of s, hence all labels are of this type.

Now, it is easy to see, that from a p-colouring with labels ai ∈ {0, . . . , p − 1} ai we can pass to a Dp colouring with labels sr ∈ Dp and vice versa.

9 3.5 The Alexander polynomial We describe the recipe to construct the Alexander polynomial using matrices similar to the colouring matrix Col(DK ). Let DK be a knot diagram

(i) label the arcs by variables xi and the crossings by cj.

(ii) Build an n × n- matrix A(DK ) by inserting for a right handed crossing cj xd

xb xa in the j-th row the entry a) (1 − t) in the a-th row, b) −1 in the b-th row, c) t in the d-th row.

xd

xa For a left handed crossing xb in the j-th row the entry a) (1 − t) in the a-th row, b) t in the b-th row, c) −1 in the d-th row.

(iii) Remove an arbitrary row and column to get a matrix Ae(DK )

k (iv) Compute detAe(DK ) and normalize it: Multiply with a factor ±t , k ∈ Z, such that the constant term of the polynomial is positive (in particular non-zero). This polynomial is the Alexander-polynomial AK (t) ∈ Z[t].

Theorem 3.13. The polynomial AK (t) is a knot invariant. For a detailed proof as given in the lecture, see [MS06]

Corollary 3.14. The value AK (−1) ∈ Z is a knot invariant, called the deter- minant of the knot. A knot is p-colourable if and only if AK (−1) is divisible by p.

See [Liv93, Exercise 5.2].

10 Properties of AK (t) (i) Alexander polynomial gives a bound on the crossing number:

degAK (t) < c(K)

(ii) it distinguishes the (2, n)-torus knots

(iii) it is multiplicative with respect to connected sum:

AK1#K2 (t) = AK1 (t) · AK2 (t), (3.4)

(iv) it is invariant under mirroring and orientation reversal,

AK = AK ,AK∗ = AK (3.5)

There are many other ways to define the polynomial AK (t), f.e. via Seifert surfaces, or via the knot group, see [Liv93]. Further Reading 3.15. For a much more conceptual understanding of the Alexander polynomial, see [Rol76].

3.6 The Alexander polynomial via skein relation Another convenient normalization of the Alexander polynomial is the normal- ization of the Conway-Alexander polynomial ∆K (z) ∈ Z[z]. To obtain this k 1 Z polynomial from the AK (t), one first finds a factor ±t , k ∈ 2 , such that k −1 ±t · AK (t) =: AgK (t) satisfies AgK (t) = AgK (t ). This can be written as a poly- 1 − 1 nomial in z = t 2 − t 2 , which is then ∆K (z). ∆K (z) satisfies a skein relation: If we consider a single crossing of K, for example L+ = ! we can draw two other links by just changing this crossing locally: L− = " and Ls = ­ Note that in general, the two other diagrams are not knots anymore, but might have several components: Definition 3.16. A link with n components is an embedding of S1 × ... × S1 → | {z } n R3, considered up to isotopy. Theorem 3.17. The Conway-Alexander polynomial is defined for links L and satisfies at each crossing the Conway skein relation (C-S)

∆L+ (z) − ∆L− (z) = −z∆Ls . (3.6) To compute the Conway-Alexander polynomial one needs to identify crossings that simplify the knot once they are reversed.

11 3.7 The Kauffman bracket and the Jones polynomial

The crossings in unoriented links have two resolvements: For L = 0 we have L1 = H and L2 = 1. Suppose we look for a polynomial < L >∈ Z[a±1, b±1, c±1] such that

(i) < L >= a < L1 > +b < L2 >, where L, L1 and L2 are as above. (ii) < L ∪ d > c· < L >, where d denotes the unknot, then, see f.e. [CDM12, Sec.2.4]: Proposition 3.18. The assignment < − > is invariant under the unoriented Reidemeister move ΩII if and only if b = a−1, c = −a2 − a−2. In this case, ΩIII follows. The polynomial < − > with the normalization < Unknot >= 1 is called the Kauffman bracket. The resulting Kauffman skein relations (K-S) read:

−1 (i) < L >= a < L1 > +a < L2 >. (ii) < L ∪ d > (−a2 − a−2)· < L >,

(iii) < d >= 1. These allow to evaluate every knot diagram. The result is independent of the order in which we resolve the crossings, since there exists a closed formula, see below. Note, that < − > is not a knot invariant: It is not invariant under ΩI: Lemma 3.19. The value of the Kauffman bracket on a right handed twist (the diagram on the left of ΩI with a right handed crossing ) is −a−3 times the value of the relsolved diagram. The value of the Kauffman bracket on a left handed twist is −a3 times the value of the relsolved diagram. Definition 3.20. The writhe w(L) of a link diagram L is the number of right handed crossings minus the number of left handed crossings in L. Definition 3.21. The Jones polynomial V (L) ∈ Z[a±1] is defined as

V (L) = (−a)−3w(L) < L > . (3.7)

− 1 A common normalization is to set a = t 4 and to regard it as polynomial J(t) − 1 in t, i.e. V (t 4 ) = J(t). Theorem 3.22. V (L) is a knot invariant. To see explicitly, that V (L) is well-defined, i.e. independent of the order in which we resolve the crossings, one proves a closed formula for V (L):

12 State sum formula See [Kau87]:

(i)A state s on an unoriented link diagram L is an assignment of ±1 to each crossing of L.

(ii) for a state s of L, define s(L) to be the collection of circles that arises from

+1 −1 the rule that for we resolve the crossing as H, while for we resolve the crossing as 1.

(iii) denote by Σs the total sum of the signs and by |s(L)| the number of circles of s(L).

Proposition 3.23. The Kauffman bracket can be computed as P < L >= X < L|s >, with < L|s >= a s(−a2 − a−2)|s(L)|−1 (3.8) s

Properties of the Jones polynomial

• The Jones polynomial can potentially detect mirror images of knots:

Proposition 3.24. If we mirror a link L, i.e. L 7→ L, the polynomials change as

< L > (a) =< L > (a−1) and V (L)(a) = V (L)(a−1). (3.9)

For example J(t) can distinguish the left from the right trefoil.

• The Jones polynom satisfies the skein relation (J-S): (i) √ 1 t−1J(!) − tJ(") = ( t − √ J(­) (3.10) t (ii) J(Unknot) = 1

3.8 The Tait conjecture We use the Jones polynomial to prove the Tait conjecture along the lines of [Kau87].

Theorem 3.25 (Tait conjecture). A reduced alternating diagram for a knot K has the minimal number of crossings among all diagrams for K.

13 Note that a given knot might not have an alternating diagram. We prove this theorem in two steps. First we show that the every reduced alternating diagram for K has the same number of crossings. Then we show that every other diagram has more or equal crossings than a reduced alternating diagram. To show the first statement we define

Definition 3.26. The span span(K) ∈ Z of a knot K is the difference of the maximal degree minus the minimal degree of the for any knot diagram DK for K, i.e.

span(K) = maxdeg(< DK >) − mindeg(< DK >) (3.11)

Lemma 3.27. The span is a well-defined knot invariant.

Proof. The difference between the Kauffman bracket and the Jones polynomial is just the writhe correction. This does not change the difference between the highest and lowest degree, thus

span(K) = maxdeg(V (K)) − mindeg(V (K)). (3.12)

Since the Jones polynomial is an invariant, it follows that span(K) is a knot invariant and well-defined, i.e. independent of which diagram DK we use in its definition. We prove

Theorem 3.28. The number #cr(DK ) of crossings in all reduced alternating diagram DK for a given knot K are the same. It satisfies

span(K) = 4 · #cr(DK ).

The second step is

Proposition 3.29. For every knot K and every knot diagram DK , the number of crossings #cr(DK ) of DK satisfies

4 · #cr(DK ) ≥ span(K). (3.13)

Hence the Tait conjecture follows, since for a reduced alternating diagram DK 0 and every other diagram DK for K we have the inequality 1 #cr(D ) = span(K) ≤ #cr(D0 ). (3.14) K 4 K

14 4 Quantum invariants

In this section we take a change of perspective: We regard a knot diagram as a “blueprint for a 2-dimensional computation”. To compute its value, we need to assign algebraic structures to few building blocks and can then glue these together to a knot invariant. We give a detailed introduction to the diagrammatic calculus, then we define tangles and specify what we mean by a quantum invariant. We show how to recouver the Kauffman bracket from the 2-dimensional calculus. Note: The actual construction of quantum invariants is only outlined in the last lecture.

4.1 Diagrammatic calculus 1-dimensional diagrammatic calculus Let X,Y,Z be sets and f : X → Y and g : Y → Z maps. We represent the sets on directed lines as

Y

X f

and the maps as labelled dots (or boxes) as X (4.1)

Z g

Then the composition f ◦ g has the graphical expression Y f X Precisely:

X n f ... n X3 • A 1-dimensional diagram is a diagram of the form f2 where the target X2 f1 X1

of fi, which is the set Xi+1 is also the source of fi+1.

• the evaluation of the diagram is the composite of the maps fn ◦ ... ◦ f2 ◦ f1, which is a map from X1 to Xn.

15 2-dimensional diagrammatic calculus In the 1-dimensional calculus we can also use vector spaces Xi and linear maps fi. In order to depict then the tensor

Y Y 0

f f 0 product of vector spaces, we need the second dimension: X X0

In this diagram, f : X → Y and f 0 : X0 → Y 0 are linear maps and the evaluation of the diagram is by definition the tensor product f ⊗ f 0 : X ⊗ X0 → Y ⊗ Y 0. Precisely: Definition 4.1. A progressive 2-d diagram consists of lines and boxes in the cube [0, 1]2, such that the projection to the y-axis is regular on each line (it is forbidden, that the lines have maxima and minima). Moreover, the boxes are connected to the lines. The lines are labelled with vector spaces and the boxes

Y1 Y2 ... Ym

T with tensors, such as corresponds to a tensor T : X1⊗X2⊗...⊗Xn → X1 X2... Xn

Y1 ⊗ Y2 ⊗ ... ⊗ Ym. The evaluation of a progressive 2d diagram goes in two steps. First step is to project the diagram to the y-axis via π : [0, 1]2 → [0, 1] to produce a 1d diagram. The labels on a point p on the y-axis correspond thereby to the tensor product of the labels on the the ordered set π−1(p). We are allowed to deform progressive diagrams by progessive isotopies φ : I × [0, 1]2 → [0, 1]2 where for each t ∈ I = [0, 1], φ(t, −) applied to the diagram is again progressive. Proposition 4.2. The evaluation of 2d diagrams is invariant under progressive isotopies. The basic move, that happens during a progressive isotopy is the move from

Y Y 0 Y Y 0 f 0 f

f to f 0 (4.2) X0 X X X0

It is clear, that the evaluation remains the same before and after the move.

16 non-progressive diagrams To evaluate non-progressive diagrams, we need to interpret maxima Ú and minima Þ in the graphical calculus. For a finite dimensional vector space there are candidates.

Definition 4.3. For a finite dimensional k-vector space V , the evaluation map ∗ ∗ evV : V ⊗ V → k, where V = Homk(V, k) is the dual vector space, is defined ∗ by evV (v ⊗ α) = α(v) with v ∈ V and α ∈ V . ∗ The coevaluation map coevV : k → V ⊗ V is defined as the unique linear map P i i that sends 1 ∈ k to coevV (1) = i e ⊗ ei. Here {ei}i is a basis of V and {e }i the corresponding dual basis of V ∗.

One easily sees, that the coevaluation map coevV is well-defined, i.e. indepen- dent of the chosen basis.

Proposition 4.4. The evaluation and coevaluation maps satisfy

(evV ⊗1) ◦ (1 ⊗ coevV ) = 1V and (1 ⊗ evV ) ◦ (coevV ⊗1) = 1V ∗ . (4.3)

These are called the snake identities .

Drawing the snake identities diagrammatically motivates to draw the evalu- ation map as evV = Þ and coevV = Ú, where in both cases, the line is labelled with V . As two applications we see that

• the graphical calculus allows to present the dual of a linear map f : V → W using the evaluation and coevaluation maps,

• using further, that the tensor product is symmetric, i.e. V ⊗ W ' W ⊗ V , it allows to also express the trace of an endomorphism f : V → V .

4.2 The Kauffman bracket from the diagrammatic calculus In order to recouver the bracket polynomials from the 2d diagrammatic calculus, there are two changes in the framework we need to take into account:

• We need unoriented lines, hence there is no canonical choice for Ù and Ü.

• As result we want a polynomial in R = C[a, a−1], hence we need to work over this ring instead of over a field.

The recipe goes as follows.

17 • Consider the 2-dimensional R-module V = Re1 ⊕ Re2 with basis e1 and e2. Then V ⊗ V has as basis e11 = e1 ⊗ e1, e12 = e1 ⊗ e2, e21 = e2 ⊗ e1 and e22 = e2 ⊗ e2. • define linear maps   F(Ù): V ⊗ V → R via F(Ù) = 0 a −a−1 0 ,  0   −a  Ü Ü   F( ): R → V ⊗ V via F( ) =  −1  ,  a  0 (4.4)  a 0 0 0   0 a−1 0  / /   F( ): V ⊗ V → V ⊗ V via F( ) =  −1 −3  .  0 a a − a 0  0 0 0 a

Proposition 4.5. By assigning these data to a 2d-diagram, the (unoriented) snake identities are satisfied. Furthermore, the assignment F satisfies the Kauff- man skein relation from Subsection 3.7.

However, the value on the unknot is F(d) = −a2 − a−2, it follows that if we devide F by this value, we would get the bracket polynomial, if we knew, that we could evaluate any unoriented knot using F. Evaluating a knot using 2d diagrammatic caluclus is the definition of a quantum invariant.

4.3 Tangles and quantum invariants To define properly, what a quantum invariant is, we first define tangles.

Definition 4.6. Let k, l ∈ N0. An (unoriented) tangle T of type (k, l) is a finite number of disjoint embedded arcs and circles in R2 × I, I = [0, 1], such that the boundary ∂T satisfies

∂T = T ∩ R2 × I = (1, 2, . . . , k × 0 × 0) ∪ (1, 2, . . . , l × 0 × 1), (4.5) and T meets its boundary transversally. Here, k = {1, 2, . . . , k}. Two tangles T and T 0 are considered equivalent, if they are related by an ambient isotopy of R2 × I that fixes the boundary.

An oriented tangle T of type (~k,~l is defined similarly, but here ~k = (+, −, −,..., −) is a sequence of signs ± of length k and similarly l. T con- sists of oriented arcs and circles and it is required to be outwards oriented at a boundary with value − and inwards oriented at a boundary of value +. Tangles can be composed in two ways:

18 • The vertical composite of an (k, l) tangle T with an (l, m) tangle T 0 is the (k, m) tangle T 0◦T , that is obtained by stacking T 0 on top of T and rescaling to end again in R2 × I. For oriented tangles, this operation is defined if ~l for T is ~l0 for T 0, where (−) denotes the reversal of the signs, i.e. + = − and − = +, applied to all entries seperately.

• The horizontal composite of an (k, l)-tangle T with an (k0, l0)-tangle T 0 is the (k + k0, l + l0)-tangle T ⊗ T 0, that is obtained by placing T 0 to the right of T . This is defined analogously for oriented tangles, here the sequences of signs get concatenated.

Theorem 4.7 ([Tur10]). Every oriented tangle diagram can be obtained by a finite vertical and horizontal composition of the following elementary tangles:

Ó, Ö, Ø, Ú, Û, Ý, Þ, (4.6)

Two tangles are equivalent, if and only if they are related by a finite seqence of the following moves (called Turaev moves):

(i) Moves as in (4.2),

(ii) Snake identities,

(iii) Turaev move T(1)

Figure 4.1: Turaev move I, image from [CDM12]

(iv) Turaev move T(2)

19 Figure 4.2: Turaev move II, image from [CDM12]

(v) Reidemeister moves ΩI, ΩII and ΩIII.

Remark 4.8. More conceptually the content of this theorem can be captured as follows: Tangles form a higher algebraic structure: There are two types of compositions and not all compositions are defined for all tangles. The structure that is present is that of a monoidal category. The theorem above then gives a presentation of this category in terms of generators and relations.

Fix a commutative ring R.

Definition 4.9. A (unframed) quantum invariant F of tangles is a R-module V together with R-linear maps

F(Ö): V ⊗ V → V ⊗ V, F(Ø): V ⊗ V → V ⊗ V, F(Ý): R → V ∗ ⊗ V, F(Þ): R → V ⊗ V ∗, (4.7) F(Ú): V ∗ ⊗ V → R F(Û)V ⊗ V ∗ → R, such that the Turaev moves are satisfied.

Note that the second Reidemeister moves is equivalent to the requirement, that the map F(Ö) is invertible with inverse F(Ø).

Remark 4.10. Following the previous remark, this definition can be stated in a more conceptual way as defining a quantum invariant as a certain type of functor, namely as a braided monoidal functor. Then it is a consequence of the theorem above, that this is equivalent to the given definition.

The tangles contain in particular the braid group.

Definition 4.11. A braid on n strands is a progressive (n, n)-tangle without circles.

Recall, that progressive means that the maxima and minima are forbidden for the tangles. Hence, braids are constructed only out of the first three elementary tangles.

20 Lemma 4.12. With the vertical composition, braids on n strands form a group Bn. Examples of quantum invariants that go beyond the Jones polynomial are outlined in the last lecture (not a content of these notes). They use certain Lie algebras.

5 Finite type invariants

Knot invariants are in a precise sense dual to knots. It turns out, that quite a lot can be said about the class of finite type invariants. In particular we will see how Lie algebras give rise to knot invariants and thus approach quantum invariants from a different perspective. Finite type invariants occured, when Vassiliev studied singular knots, i.e. smooth maps φ : S1 → R3 that fail to be embeddings. The simplest case is: Definition 5.1. A double point of φ : S1 → R3 is a point p ∈ im(φ), such that φ−1(p) consists of precisely two points, that meet transversaly at p. Let R be some abelian group and ν : K → R a knot invariant (recall, that K is the set of isotopy classes of oriented knots). The extension of ν to knots with a double point is defined via the Vassilev skein relation (V-S):

ν( ) = ν(!) − ν("). (5.1)

On knots with n double points, ν is extended by applying (V-S) to all double points. Definition 5.2. A knot invariant ν : K → R is a Vassilev invariant of order ≤ n, if its extension to singular knots with strictly more than n double points vanishes. ν is of order n, if it is of order ≤ n, but not of order ≤ n − 1. Put differently, for a Vassiliev invariant ν of order n, there exists a knot K with n double points and ν(K) 6= 0. Denote by Vn the set of Vassiliev invariants of order ≤ n. By definition, Vn ⊂ Vn+1. Hence, there is a filtration

∞ [ V0 ⊂ V1 ⊂ ... ⊂ V := Vn (5.2) n=0

0 1 n Write the Conway-Alexander polynomial as ∇K (z) = ∇K + ∇K · z + ... + ∇K · zn + ...

n Lemma 5.3. The nth coefficient of the Conway-Alexander polynomial ∇K is a Vassiliev invariant with values in Z of order ≤ n.

21 Analogy to polynomials Finite type invariants are in certain respects analogous to polynomials. Let ν be a knot invariant. • Call Dnν its extension to knots with n double points via (V-S). Then D is an operator that turns Dnν into Dn+1ν. Now, ν is a Vassiliev invariant of order ≤ n if Dn+1ν = 0. Analogously, a an- d n+1 alytic function f(x) is a polynomial of degree ≤ n if and only if ( dx ) f = 0

• If n1 and n2 are invariants with values in a commutative ring R, we can define their pointwise product ν1 ·ν1, that takes values (ν1 ·ν2)(K) = ν1(K)· ν2(K). If ν1 and ν2 are Vassiliev invariants of order ≤ n1 and ≤ n2, then ν1 · ν2 is a Vassiliev invariant of order ≤ (n1 + n2). • One of the main questions in the field is to which extend a given knot invariant can be approximated by Vassiliev invariants, similar to the ap- proximation of analytic functions by polynomials.

The spaces Vn

•V 0 is 1-dimensional: A ν ∈ V0 vanishes on all knots with a double point, hence ν( ) = ν(!) − 0 (5.3) Hence, ν does not chance if we flip a crossing. Hence its value is the same on every knot.

•V 1 is 1-dimensional: Let K be a knot with one double point. If ν ∈ V1, we can again flip all crossings in K without changing the value of ν. By that we can transform K into the singular knot that looks like the symbol “8”. Its value on this knot is 0 by (V-S). Hence it follows, that ν ∈ V0.

Proposition 5.4. The dimension of V2 is 2 over R.

Proof. For the proof, we consider a knot K with two double points p1 and p2. When travelling along K there are two possibilities: In case 1 we obtain the sequence (p1p1p2p2) or a cyclic permutation thereof (if we choose a different start point, the sequence permuts cyclically). In case 2 we obtain (p1p2p1p2) or a cyclic permutation thereof. Again, the value of ν ∈ V2 does not change if we flip a crossing in K, but by flipping crossings we can not change the sequences in case 1 and 2. Hence we can pick two standard knots with two double points K1 and K2 that represent the cases 1 and 2, and evaluate n on these knots. This defines 2 a map α2 : V2 → R . The kernel of α2 consists of all invariants, that vanish on all knots with two dou- ble points, that is ker(α2) = V1. It can be shown, that its image is 1-dimensional, a generator consists of ∇2, the second coefficient of the Conway-Alexander poly- nomial. The statement follows.

22 Combinatorial knot diagrams To reveal the structure that is present in the previous proposition, we use chord diagrams. First, a diagrammatic notation for conventional knots. All diagrams are considered up to diffeomorphisms of the circle. Definition 5.5. A Gauss diagram is an oriented circle with a set of ordered pairs of distinct points. Each pair of points carries a sign in {±}. We obtain an injective map from knot diagrams up to ambient isotopy to Gauss diagrams by travelling along a knot diagram and recording the two points where the crossings take place. The points are ordered from the over- to the undercrossing (taken into account by connecting the points with an arrow) and they carry a + for a right handed crossing and a − for a left handed crossing. Note, that not every Gauss diagram comes from a knot. Definition 5.6. A chord diagram of order n is an oriented circle with a set of n disjoint (unordered) pairs of points, the chords. The set of chord diagrams of order n is denoted An. We depict a chord diagram by joining the pairs of points with an unoriented arc in the circle. The arc is also called chord. The chord diagram σ(K) ∈ An of a singular knot K with n double points is obtained by marking the double points on K and travelling across K. Every chord diagram D comes from a singular knot, called a realisation of D. Analogously to the discussion of V2, we obtain Proposition 5.7. The value of a Vassiliev invariant ν of order ≤ n on a knot K with n double points depends only on the chord diagram of K: If K1 and K2 are two knots with n double points and σ(K1) = σ(K2), then ν(K1) = ν(K2).

Thus, with RAn := F un(An,R) the set of functions from An to R, we obtain a well-defined map

αn : Vn → RAn, with αn(ν)(D) = ν(K), (5.4) where D ∈ An and K is any knot with σ(K) = D. From the definitions we conclude

Lemma 5.8. The kernel of αn is Vn−1.

Vn Hence, αn induces an injective map αn : → RAn. Since the set An is Vn−1 finite, it follows, that also Vn is finitely generated over R and thus, inductively, Vn−1 also Vn is finitely generated over R. Next we describe the image of α. We say that a chord diagram D ∈ An has an isolated chord, if there exists a chord in D, that does not intersect any other chord in D.

23 Definition 5.9. A function f ∈ RAn satisfies the 4T-relation, if

Figure 5.1: 4T -relation, image from [CDM12]

Such an f is also called a framed weight system. f satisfies the 1T-relation, if it vanishes on chord diagrams with isolated chords. A function f satisfying the 4T- and 1T-relations is called an unframed weight system. The set of these is denoted Wn. The fundamental theorem of Vassiliev invariants:

Vn Theorem 5.10. The image of αn : → RAn consists precisely of the un- Vn−1 framed weight systems Wn. The hard part is to show surjectivity. Here, the Kontsevich integral provides an explicit right inverse to αn.

Further examples of finite type invariants The Jones polynomial JK (t) pro- vides finite type invariants. If we write t = eh with a new variable h and consider the function 0 1 n n JK (h) = JK + JK · h + ... + JK · h + ·,

n Lemma 5.11. The coefficient JK is a Vassiliev invariant of order ≤ n. Next we consider an important class of framed weight systems that provide Vassiliev invariants according to the fundamental theorem. Recall the following pertinent definition:

Definition 5.12. A Lie algebra over C is a C-vector space g togther with a bilinear map [−, −]: g × g → g, the Lie bracket, such that for all elements x, y, z ∈ g,

(i) [x, y] = −[y, x],

(ii) [x, [y, z]] = [[x, y], z] + [y, [x, z]] (Jacobi-identity).

P In a basis {ei} of g, the Lie bracket defines numbers cijk with [ei, ej] = k cijkek Examples include all C-algebras A as for instance A = Matn(C) with the Lie bracket [x, y] = xy − yx for x, y ∈ A. Another example is sln = {X ∈ Matn(C)|tr(X) = 0}, again with the commu- tator [x, y] = xy − yx as Lie bracket.

Definition 5.13. Let g be a Lie algebra.

24 (i) A bilinear form < −, − >: g × g → C is called ad-invariant, if for all x, y, z ∈ g < [x, z], y >=< x, [z, y] > . (5.5)

(ii) A Lie algebra g with an ad-invariant symmetric non-degenerate bilinear form < −, − > is called metrized.

In a metrized Lie algebra, the structure constants are cylically symmetric. We furthermore need

⊗n • For any vector space V , T (V ) = ⊕nV is the tensor algebra over V with multiplication given by the tensor product.

• If g is a Lie algebra, consider the both-sided ideal < x ⊗ y − y ⊗ x − [x, y] > in T (g) generated by all elements of the form x ⊗ y − y ⊗ x − [x, y] in T (g) with x, y ∈ g. The algebra . U(g) := T (g) < x ⊗ y − y ⊗ x − [x, y] > (5.6) is called universal enveloping algebra over g.

Now we are in a position to define for every metrized Lie algebra g and every n ∈ N a framed weight system

n ϕg : An → Z(U(g)) of degree n with values in the center of the universal enveloping algebra Z(U(g)): Pick an orthonormal basis {ei} of g. To this end, first decorate the chords in a chord diagram D with variables i, j, k, . . ., choose a starting point on D and then multiply the basis elements ei, ej, ek in U(g) according to their occurence on the chord and finally sum the variables over the basis of g. The result is n ϕg (D) ∈ U(g) Theorem 5.14. Let g be a metrized Lie algebra and n ∈ N.

n (i) The element ϕg (D) for D ∈ An depends not the chosen basis or the starting point in D.

n (ii) It lands in the center, i.e. ϕg (D) ∈ Z(U(g)).

n (iii) The function ϕg : An → Z(U(g)) satisfies the 4T-relation.

n Hence, ϕg is a framed weight system of degree n and thus corresponds to a framed Vassiliev invariant. If we use as Lie algebra sl2, we recouver by this procedure the Jones polynomial. We will briefly discuss framed knot invariants and quantum invariants associ- ated with metrized Lie algebras in the last lecture. It turns out, that also from a

25 Quantum Chern Simons theory for g ←→ framed weight systems ϕg ←→ Quantum invariants for g physical perspective, metrized Lie algebras arise as the Lie algebras of the gauge group in Chern-Simons theory. Its quantization physically motivates the exis- tence of knot invariants associated with such Lie algebras. Thus we arrive at three descriptions of the same important class of knot invariants with the Jones polynomial as most prominent example:

Further Reading 5.15. The quantum Chern Simons theory is mathematically described as Reshethikin-Turaev theory, [RT91]. In [RT91] it is constructed for the lie algebra su2, for the general case see [BKJ01]. The passage from quantum Chern Simons theory to Vassiliev invariants is explained in Dror Bar-Natans thesis, [BN91]. The relation between weight systems and quantum invariants is outlined in [CDM12, Sec. 11.2.2].

6 References

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[BN91] D. Bar-Natan. Perturbative aspects of the chern-simons topological quantum field theory. Ph. D. Thesis, 1:151, 1991.

[CDM12] S. Chmutov, S. Duzhin, and J. Mostovoy. Introduction to Vassiliev knot invariants. Cambridge University Press, 2012.

[Kau87] L. H. Kauffman. State models and the jones polynomial. Topology, 26(3):395–407, 1987.

[Liv93] C. Livingston. Knot theory, volume 24. Cambridge University Press, 1993.

[MS06] R. Messer and P. Straffin. Topology now! MAA, 2006.

[Rol76] D. Rolfsen. Knots and links, volume 346. American Mathematical Soc., 1976.

[RT91] N. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103(3):547–597, 1991.

26 [Saw96] S. Sawin. Links, quantum groups and tqfts. Bulletin of the American Mathematical Society, 33(4):413–445, 1996.

[Sul00] M. C. Sullivan. Knot factoring. The American Mathematical Monthly, 107(4):297–315, 2000.

[Tur10] V. G. Turaev. Quantum invariants of knots and 3-manifolds. 2nd re- vised ed. de Gruyter Studies in Mathematics, 2010.

[Wit89] E. Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys., 121(3):351–399, 1989.

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