Introduction to knot theory

Transcription

1 Introduction to knot theory Summary of the lecture by Gregor Schaumann 2016 Fakultät für Mathematik, Universität Wien, Austria This is a short summary of the lecture Introduction to knot theory, 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

2 Introduction Knots are topological objects familiar from daily experience. Two natural questions arise: (i) What is a good mathematical concept of a knot? considered to be equivalent? When are two knots (ii) Can one classify all possible knots? Can one distinguish knots? It turns out, that classifying knots is very hard, knot theory focusses mainly on tools to distinguish knots, so called knot invariants. In modern physics, knots arise as worldlines of quantum particles in a 3 dimensional 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 Jones polynomial 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 introduced and its relation to the knot colourings is discussed. Finally, an alternative definition of the Alexander polynomial using the Conway-Alexander skein relation 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

3 to time reasons, but instead use an explicit presentation of tangles via generators and relations. Finally, Vassiliev invariants provide another systematic treatment of knot invariants. 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 : S 1 S 1 = {(x, y) R 2 x 2 + y 2 = 1} is the circle. R 3, where We always the orientation of circle that is counter-clockwise. This gives also an orientation to every parametrized knot. A reparametrization φ of a parametrized knot φ is an orientation preserving diffeomorphism diffeomorphism f : S 1 S 1 such that φ = 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 φ 1 and φ 2 be two parametrized knots. An isotopy F : φ 1 φ 2 is a smooth map F : S 1 [0, 1] R 3 such that (i) F (x, 0) = φ 1 (x) for all x S 1, (ii) F (x, 1) = φ 2 (x) for all x S 1, (iii) F (, u) : S 1 R 3 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

4 1.2 Knot diagrams Definition 1.6. A knot diagram for an oriented knot φ is a plane P R 3 with a projection π : R 3 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 neighbourhood of a single crossing. 1.3 Operations on knots Denote by σ S 1 : S 1 S 1 the orientation reversal of S 1 and by τ R 3 : R 3 R 3 the reflection along some plane in R 3. This defines two operations on oriented knots φ : S 1 R 3 : (i) orientation reversal φ φ σ S 1, (ii) mirror φ τ R 3 φ. This is well defined on equivalence classes K K and defines operation K 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 σ 2 = 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 K 1 #K 2 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

5 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 = S 1 S 1 the torus. The (p,q)-torus knot for p, q two coprime integers is the embedding S 1 T 2 that winds p times around the meridian (first S 1 ) of T 2 and q times around the longitude (second S 1 ) of T 2. Lemma 2.1. Every embedding of S 1 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 Seifert surface for an oriented knot K is an embedded oriented surface F in R 3 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(f K ) among all Seifert surfaces F K 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(k 1 #K 2 ) = g(k 1 ) + g(k 2 ). 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 dimensions, see [Rol76] 3 Invariants of knots Definition 3.1. Let S be a set. A knot invariant ν with values in S is a function ν : K S. Main principle: If for an invariant ν, ν(k 1 ) ν(k 2 ), then the knots K 1 and K 2 are not equivalent. 5

6 3.1 First examples The genus is an invariant g : K N 0, The crossing number c : K N 0 is the minimal number of crossings among all diagrams for a given knot. The unknotting number u : K N 0 is the minimal number of crossings that is required to change any diagram of a given knot into a diagram for the unknot. The relation u(k) c(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 D are called ambient isotopic, if there exists a smooth map F : R 2 [0, 1] R 2 with F (, 0) = id and F (, 1)(D) = D. 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 Col 3 = {Red, Blue, Y ellow}. We call an arc in a knot diagram a segment of the diagram from an undercrossing to an undercrossing. 6

7 Definition 3.3. A knot diagram D K for a knot K is called colourable, if each arc of D K can be coloured using the three colours of Col 3, 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 Reidemeister 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 N 0. 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 Col 3 {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 such that x = z p + q) (mod p) means that there exists z Z We immediately generalize this to p-colourability, with Col p = {0, 1,..., p 1}: Definition 3.7. A knot diagram D K for a knot K is called colourable mod p, or p-colourable for a prime p > 2, if each arc of D K can be labelled with elements in Col p, such that (i) at least two numbers are used, 7

8 (ii) at any crossing where the overcrossing arc is coloured with x, the two other arcs with y and z, the condition is satisfied. 2x y z 0 (mod p) (3.2) Theorem 3.8. Being colourable mod p or not is a knot invariant. Col p : K {±}. If a knot K is colourable, the value of the invariant is Col p (K) = +, otherwise Col p (K) =. Again one can show, that the number of possible p-colourings is a knot invariant. 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 D K is colourable mod p, we (i) label each arc with a variable x i, (ii) label the crossings with c i, (iii) the relations 2x i x j x k 0 mod p on each crossing define an n n- matrix Col(D K) with columns the variables x i and rows the c j. 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 D K is colourable mod p if and only if the equation Col(D K ) x 0 mod p (3.3) has a solution with x i {0, 1,..., p 1} and at least two x i different. We will show in the next section, that Lemma 3.9. (i) Any of the rows and columns of Col(D K ) is a linear combination of all the others with non-zero coefficients. If we erase an arbitrary row and column we obtain an (n 1) (n 1)-matrix Col tunc (D K ), the truncated colouring matrix. (ii) The knot K can be coloured mod p if and only if Col trunc (D K ) x 0 mod p, which is the case if and only if det(col trunc (D K ) is divisible by p. 8

9 knot group We introduce the knot group as the universal colouring group. First we generalize p-colourability further. Definition 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 (ii) at each right handed crossing k g h, gkg 1 = h, at each left handed crossing, h g, ghg 1 = k. We can turn this around and define a group G(D K ) for each knot diagram D K by assigning generators to the arcs of D K and the relations of Definition 3.10 ii) for all crossings. Theorem The group G(D K ) is isomorphic to the fundamental group of R 3 \ 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. We find again p-colourability by using the dihedral group D p =< r, s r p, s 2, srsr >. Lemma Let D K be a knot diagram that is coloured by D p. Then all labels on the arcs of the diagram are of the form sr a i with a i {0,..., p 1}. Proof. The elements of D p can be brought into a normal form of the type s ɛ r a i with ɛ {0, 1} and a i {0,..., p 1}. It is easy to see, that the elements of the form sr a i are invariant under conjugation, i.e. x sr ai 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 sr a i, 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 a i {0,..., p 1} we can pass to a D p colouring with labels sr a i D p and vice versa. 9

10 3.5 The Alexander polynomial We describe the recipe to construct the Alexander polynomial using matrices similar to the colouring matrix Col(D K ). Let D K be a knot diagram (i) label the arcs by variables x i and the crossings by c j. (ii) Build an n n- matrix A(D K ) by inserting for a right handed crossing c j x d x b x a 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. x d For a left handed crossing x b x a 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 Ã(D K) (iv) Compute detã(d K) and normalize it: Multiply with a factor ±t k, k Z, such that the constant term of the polynomial is positive (in particular non-zero). This polynomial is the Alexander-polynomial A K (t) Z[t]. Theorem The polynomial A K (t) is a knot invariant. For a detailed proof as given in the lecture, see [MS06] Corollary The value A K ( 1) Z is a knot invariant, called the determinant of the knot. A knot is p-colourable if and only if A K ( 1) is divisible by p. See [Liv93, Exercise 5.2]. 10

11 Properties of A K (t) (i) Alexander polynomial gives a bound on the crossing number: dega K (t) < c(k) (ii) it distinguishes the (2, n)-torus knots (iii) it is multiplicative with respect to connected sum: A K1 #K 2 (t) = A K1 (t) A K2 (t), (3.4) (iv) it is invariant under mirroring and orientation reversal, A K = A K, A K = A K (3.5) There are many other ways to define the polynomial A K (t), f.e. surfaces, or via the knot group, see [Liv93]. via Seifert Further Reading 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 normalization of the Conway-Alexander polynomial K (z) Z[z]. To obtain this polynomial from the A K (t), one first finds a factor ±t k, k 1 Z, such that 2 ±t k A K (t) =: ÃK(t) satisfies ÃK(t) = ÃK(t 1 ). This can be written as a polynomial in z = t 1 2 t 1 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 L s = Note that in general, the two other diagrams are not knots anymore, but might have several components: Definition A link with n components is an embedding of S 1... S 1 }{{} n R 3, considered up to isotopy. Theorem 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

12 3.7 The Kauffman bracket and the Jones polynomial The crossings in unoriented links have two resolvements: For L = 0 we have L 1 = H and L 2 = 1. Suppose we look for a polynomial < L > Z[a ±1, b ±1, c ±1 ] such that (i) < L >= a < L 1 > +b < L 2 >, where L, L 1 and L 2 are as above. (ii) < L d > c < L >, where d denotes the unknot, then, see f.e. [CDM12, Sec.2.4]: Proposition The assignment < > is invariant under the unoriented Reidemeister move ΩII if and only if b = a 1, c = a 2 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: (i) < L >= a < L 1 > +a 1 < L 2 >. (ii) < L d > ( a 2 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 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 a 3 times the value of the relsolved diagram. Definition 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 The Jones polynomial V (L) Z[a ±1 ] is defined as V (L) = ( a) 3w(L) < L >. (3.7) A common normalization is to set a = t 1 4 in t, i.e. V (t 1 4 ) = J(t). and to regard it as polynomial J(t) Theorem 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

13 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 the rule that for +1 we resolve the crossing ash, while for 1 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 The Kauffman bracket can be computed as < L >= s < L s >, with < L s >= a s ( a 2 a 2 ) s(l) 1 (3.8) Properties of the Jones polynomial The Jones polynomial can potentially detect mirror images of knots: Proposition If we mirror a link L, i.e. L 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) (ii) J(Unknot) = The Tait conjecture t 1 J(!) tj(") = ( t 1 t J( ) (3.10) 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

14 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 The span span(k) Z of a knot K is the difference of the maximal degree minus the minimal degree of the bracket polynomial for any knot diagram D K for K, i.e. span(k) = maxdeg(< D K >) mindeg(< D K >) (3.11) Lemma 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 D K we use in its definition. We prove Theorem The number #cr(d K ) of crossings in all reduced alternating diagram D K for a given knot K are the same. It satisfies The second step is span(k) = 4 #cr(d K ). Proposition For every knot K and every knot diagram D K, the number of crossings #cr(d K ) of D K satisfies 4 #cr(d K ) span(k). (3.13) Hence the Tait conjecture follows, since for a reduced alternating diagram D K and every other diagram D K for K we have the inequality #cr(d K ) = 1 4 span(k) #cr(d K). (3.14) 14

15 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 and the maps as labelled dots (or boxes) as X f (4.1) Then the composition f g has the graphical expression Precisely: Z Y X g f A 1-dimensional diagram is a diagram of the form of f i, which is the set X i+1 is also the source of f i+1. X n f n... X 3 X 2 X 1 f 2 f 1 where the target the evaluation of the diagram is the composite of the maps f n... f 2 f 1, which is a map from X 1 to X n. 15

16 2-dimensional diagrammatic calculus In the 1-dimensional calculus we can also use vector spaces X i and linear maps f i. In order to depict then the tensor Y Y product of vector spaces, we need the second dimension: X f X f In this diagram, f : X Y and f : X Y are linear maps and the evaluation of the diagram is by definition the tensor product f f : X X Y Y. 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 Y 1 Y 2... Y m with tensors, such as Y 1 Y 2... Y m. T X 1 X 2... X n corresponds to a tensor T : X 1 X 2... X n 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 f Y Y f X f X to X X f (4.2) It is clear, that the evaluation remains the same before and after the move. 16

17 non-progressive diagrams To evaluate non-progressive diagrams, we need to Ú and minima Þ in the graphical calculus. For a finite interpret maxima dimensional vector space there are candidates. Definition 4.3. For a finite dimensional k-vector space V, the evaluation map ev V : V V k, where V = Hom k (V, k) is the dual vector space, is defined by ev V (v α) = α(v) with v V and α V. The coevaluation map coev V : k V V is defined as the unique linear map that sends 1 k to coev V (1) = i e i e i. Here {e i } i is a basis of V and {e i } i the corresponding dual basis of V. is well-defined, i.e. indepen- One easily sees, that the coevaluation map coev V dent of the chosen basis. Proposition 4.4. The evaluation and coevaluation maps satisfy (ev V 1) (1 coev V ) = 1 V and (1 ev V ) (coev V 1) = 1 V. (4.3) These are called the snake identities. Drawing the snake identities diagrammatically motivates to draw the evaluation map as ev V = Þ and coev V = 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 Ü. Ù and 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

18 Consider the 2-dimensional R-module V = Re 1 Re 2 with basis e 1 and e 2. Then V V has as basis e 11 = e 1 e 1, e 12 = e 1 e 2, e 21 = e 2 e 1 and e 22 = e 2 e 2. define linear maps F(Ù) : V V R via F(Ù) = ( 0 a a 1 0 ), 0 a F(Ü) : R V V via F(Ü) = a 1, 0 a a F(/) : V V V V via F(/) = a 1 a a a (4.4) Proposition 4.5. By assigning these data to a 2d-diagram, the (unoriented) snake identities are satisfied. Furthermore, the assignment F satisfies the Kauffman skein relation from Subsection 3.7. However, the value on the unknot is F(d) = a 2 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 N 0. An (unoriented) tangle T of type (k, l) is a finite number of disjoint embedded arcs and circles in R 2 I, I = [0, 1], such that the boundary T satisfies T = T R 2 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 are considered equivalent, if they are related by an ambient isotopy of R 2 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 consists 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

19 The vertical composite of an (k, l) tangle T with an (l, m) tangle T is the (k, m) tangle T T, that is obtained by stacking T on top of T and rescaling to end again in R 2 I. For oriented tangles, this operation is defined if l for T is l for T, 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 (k, l )-tangle T is the (k + k, l + l )-tangle T T, that is obtained by placing T 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

20 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, F(Ú) : V V R F(Û)V V R, (4.7) 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 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 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

21 Lemma With the vertical composition, braids on n strands form a group B n. 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 : S 1 R 3 that fail to be embeddings. The simplest case is: Definition 5.1. A double point of φ : S 1 R 3 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) 0. Denote by V n the set of Vassiliev invariants of order n. By definition, V n V n+1. Hence, there is a filtration V 0 V 1... V := V n (5.2) n=0 Write the Conway-Alexander polynomial as K (z) = 0 K + 1 K z n K z n +... Lemma 5.3. The nth coefficient of the Conway-Alexander polynomial n K is a Vassiliev invariant with values in Z of order n. 21

22 Analogy to polynomials Finite type invariants are in certain respects analogous to polynomials. Let ν be a knot invariant. Call D n ν its extension to knots with n double points via (V-S). Then D is an operator that turns D n ν into D n+1 ν. Now, ν is a Vassiliev invariant of order n if D n+1 ν = 0. Analogously, a analytic function f(x) is a polynomial of degree n if and only if ( d dx )n+1 f = 0 If n 1 and n 2 are invariants with values in a commutative ring R, we can define their pointwise product ν 1 ν 1, that takes values (ν 1 ν 2 )(K) = ν 1 (K) ν 2 (K). If ν 1 and ν 2 are Vassiliev invariants of order n 1 and n 2, then ν 1 ν 2 is a Vassiliev invariant of order (n 1 + n 2 ). 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 approximation of analytic functions by polynomials. The spaces V n V 0 is 1-dimensional: A ν V 0 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 V 1, 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 V 0. Proposition 5.4. The dimension of V 2 is 2 over R. Proof. For the proof, we consider a knot K with two double points p 1 and p 2. When travelling along K there are two possibilities: In case 1 we obtain the sequence (p 1 p 1 p 2 p 2 ) or a cyclic permutation thereof (if we choose a different start point, the sequence permuts cyclically). In case 2 we obtain (p 1 p 2 p 1 p 2 ) or a cyclic permutation thereof. Again, the value of ν V 2 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 K 1 and K 2 that represent the cases 1 and 2, and evaluate n on these knots. This defines a map α 2 : V 2 R 2. The kernel of α 2 consists of all invariants, that vanish on all knots with two double points, that is ker(α 2 ) = V 1. It can be shown, that its image is 1-dimensional, a generator consists of 2, the second coefficient of the Conway-Alexander polynomial. The statement follows. 22

23 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 A n. 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) A n 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 V 2, 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 K 1 and K 2 are two knots with n double points and σ(k 1 ) = σ(k 2 ), then ν(k 1 ) = ν(k 2 ). Thus, with RA n := F un(a n, R) the set of functions from A n to R, we obtain a well-defined map α n : V n RA n, with α n (ν)(d) = ν(k), (5.4) where D A n and K is any knot with σ(k) = D. From the definitions we conclude Lemma 5.8. The kernel of α n is V n 1. V Hence, α n induces an injective map α n : n V n 1 RA n. Since the set A n is finite, it follows, that also Vn V n 1 is finitely generated over R and thus, inductively, also V n is finitely generated over R. Next we describe the image of α. We say that a chord diagram D A n has an isolated chord, if there exists a chord in D, that does not intersect any other chord in D. 23

24 Definition 5.9. A function f RA n 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 W n. The fundamental theorem of Vassiliev invariants: RA n consists precisely of the un- Theorem The image of α n : framed weight systems W n. V n V n 1 The hard part is to show surjectivity. Here, the Kontsevich integral provides an explicit right inverse to α n. Further examples of finite type invariants The Jones polynomial J K (t) provides finite type invariants. If we write t = e h with a new variable h and consider the function J K (h) = JK 0 + JK 1 h JK n h n +, Lemma The coefficient J n K 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 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). In a basis {e i } of g, the Lie bracket defines numbers c ijk with [e i, e j ] = k c ijk e k Examples include all C-algebras A as for instance A = Mat n (C) with the Lie bracket [x, y] = xy yx for x, y A. Another example is sl n = {X Mat n (C) tr(x) = 0}, again with the commutator [x, y] = xy yx as Lie bracket. Definition Let g be a Lie algebra. 24

25 (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 For any vector space V, T (V ) = n V n 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 ϕ n g : A n Z(U(g)) of degree n with values in the center of the universal enveloping algebra Z(U(g)): Pick an orthonormal basis {e i } 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 e i, e j, e k in U(g) according to their occurence on the chord and finally sum the variables over the basis of g. The result is ϕ n g (D) U(g) Theorem Let g be a metrized Lie algebra and n N. (i) The element ϕ n g (D) for D A n depends not the chosen basis or the starting point in D. (ii) It lands in the center, i.e. ϕ n g (D) Z(U(g)). (iii) The function ϕ n g : A n Z(U(g)) satisfies the 4T-relation. Hence, ϕ n g is a framed weight system of degree n and thus corresponds to a framed Vassiliev invariant. If we use as Lie algebra sl 2, we recouver by this procedure the Jones polynomial. We will briefly discuss framed knot invariants and quantum invariants associated with metrized Lie algebras in the last lecture. It turns out, that also from a 25

26 Quantum Chern Simons theory for g framed weight systems ϕ g 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 existence 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 The quantum Chern Simons theory is mathematically described as Reshethikin-Turaev theory, [RT91]. In [RT91] it is constructed for the lie algebra su 2, 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 ]. 6 References [BKJ01] B. Bakalov and A. Kirillov Jr. Lectures on tensor categories and modular functors. University Lecture Series, AMS., [BN91] D. Bar-Natan. Perturbative aspects of the chern-simons topological quantum field theory. Ph. D. Thesis, 1:151, [CDM12] S. Chmutov, S. Duzhin, and J. Mostovoy. Introduction to Vassiliev knot invariants. Cambridge University Press, [Kau87] [Liv93] L. H. Kauffman. State models and the jones polynomial. Topology, 26(3): , C. Livingston. Knot theory, volume 24. Cambridge University Press, [MS06] R. Messer and P. Straffin. Topology now! MAA, [Rol76] [RT91] D. Rolfsen. Knots and links, volume 346. American Mathematical Soc., N. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103(3): ,

27 [Saw96] [Sul00] [Tur10] [Wit89] S. Sawin. Links, quantum groups and tqfts. Bulletin of the American Mathematical Society, 33(4): , M. C. Sullivan. Knot factoring. The American Mathematical Monthly, 107(4): , V. G. Turaev. Quantum invariants of knots and 3-manifolds. 2nd revised ed. de Gruyter Studies in Mathematics, E. Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys., 121(3): ,