1 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 The basic text of this lecture is [7, Chapter 1 and 2]. Link diagrams and Reidemeister’s Theorem 1. 3 3 ⊂ R We work with combinatorial links in be a combinatorial in this section. Let R L 3 v L if there exists an open neighborhood locally linear at 3 v of ∈ ∈ R v , link. We call L U 3 R ( such that L ∩ U = l ∩ U . Let V = V and an affine line l ) ⊂ L be the finite subset of ⊂ L L L is not locally linear. This finite subset of L is called the set of vertices points of where . L of 3 2 R R → Let π be the standard projection π ( Definition 1.1. ) = ( x,y ) . We say that : x,y,z is in general position with respect to π if L − 1 p π ( L ) : | π ∀ (1.1) ( p ) ∩ L |≤ 2 ∈ p ( L ) , and if moreover π while equality holds for only finitely many ∈ − 1 π ( V ) : | π (1.2) ∀ ( p ) ∩ L | = 1 . p ∈ − 1 π ( L ) is a point p ∈ π ( L ) such that the equality | A crossing of ( p ) ∩ L | = 2 holds. We call π π L ) a regular projection of L . ( Every equivalence class L ] of combinatorial links contains a representative Proposition 1.2. [ which is in general position with respect to π . L = Let be the triangulation of L whose set of 0-simplices is equal to V Proof. V ( L ). Let S v ∈ V and let σ and τ be the two 1-simplices of S containing v . There exists a small open ball ′ ′ U v 3 ∈ U we can modify L within its equivalence class to a link L such that for any by a v − 1 ′ by v while adapting σ and τ accordingly. Let -move replacing v combination of a ∆ and a ∆ a local deformation at v of us call such a modification of . Observe the special case where L L ′ in the affine line l we choose v τ . This changes the direction of σ by changing defined by τ the length (but not its direction) of τ v inside the line determined by moving its end point at τ L fixed. A deformation of L is a by , while keeping the directions of all other 1-simplices of sequence of local deformations. L is called stable if it is invariant for all sufficiently small A property of the embedding V local deformations of unstable otherwise. For example, let N = | , and ( L ) | . Consider L the number N ( L,π ) := | π ( V ( L )) | . Clearly N ( L,π ) ≤ N , and the property N ( L,π ) < N is L unstable while the property ) = N is stable. Hence for any L,π there exist arbitrarily ( N ′ ′ of L such that N ( L small deformations ,π L N . Without loss of generality we may ) = N L,π ) = N . From now on we consider only sequences of local ( therefore assume that L which do not disturb this stable property. Then the images π ( σ ) of the deformations of 2 σ of L are all 1-simplices in R 1-simplices . Let M denote the number of 1-simplices in S . π Consider the number L,π ) of distinct directions of ( ( σ ) where σ runs over the set of M 1-simplices of S . Clearly M ( L,π ) ≤ M , and is stable only if equality holds. Hence we may assume from now on that all projections π ( σ ) have distinct directions and that we are only 1

2 2 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 considering sequences of local deformations from now on which do not disturb this property. 1 − In this situation the inverse image c ) of any element c ∈ π ( L ) is finite and there exist π ( − 1 1. For ) such that | π ∈ L ( c ) | > ( v ∈ V ( L ), let b ( v ) be the c π only finitely many points of L such that π ( v ) ∈ π ( σ ). Thus b ( number of 1-simplices ) ≥ 2 for all v ∈ V ( L ). Consider σ v now the number ∑ 2) − v ) ( b ( (1.3) B ) := L,π ( L v ( V ∈ ) ( L,π ) is B ( L,π ) = 0. Let us assume this now. Observe next Clearly the only stable value of B π that the projection L ) can be deformed at any chosen vertex π ( v ) by the projection of the ( deformations as discussed above: we can move ( v ) over a sufficiently small distance in the π π ( ) while adapting π ( τ ) and π ( σ ) accordingly and keeping the remaining line determined by τ L ( ) fixed (where σ and τ are the 1-simplices of π containing v , as above). We call part of L 1 − c ∈ π ( L ) with | π c the finite set ( C ) | of points 1 the set of crossing points of π ( L ). Since > − 1 L,π ) = 0, no vertex ( v ) with v ∈ V is a crossing point. Hence the order d ( c ) := | π ( π ( c ) | B of a crossing point can be defined as the number of 1-simplices ( σ ) such that c π π ( σ ). Let ∈ c ) to be a crossing point of order d L , and apply the above type of local deformation of π ( 0 0 ). This changes the direction of π c one of the projected 1-simplices ∈ π ( σ ) such that π ( σ ) ( σ 0 in a small but arbitrary way, while keeping the directions of the lines of the projections of fixed. Then c − is replaced after this deformation by ( d the other 1-simplices in L 1) new 0 0 = 2, is reduced to d crossing points which have order 2, whereas the order of − 1 (so if d c 0 0 0 L is no longer a crossing point). Therefore, by such deformation of the original point the c 0 number ∑ (1.4) L,π ) := D Z ∈ 2) ( d ( c ) − ( ≥ 0 C c ∈ d c ) − 2 is positive, and unchanged otherwise. Hence the only stable value is reduced by 1 if ( 0 D ( L,π ) is 0. So now we have finally obtained the situation N of L,π ) − N = M ( L,π ) − M = ( B ( L,π ) = D ( L,π ) = 0, which amounts to saying that L is L regular for π . 2 Definition 1.3. is the image of a PL-map F : X → R A (PL-)link projection , where X Π F is a finite disjoint union of PL-circles, such that is injective except at finitely many double points p ∈ Π , and has the property of being locally linear at the pre-images of the double points. In particular, Π has the structure of a 4-valent planar graph (in which the 4-valent vertices are the double points of Π). 2 S X Let Π = ⊂ R ( be a link projection. Clearly there exists a triangulation F of X such ) that F is simplicial on S and such that the pre-images of the double points of Π are interior points of 1-simplices of S . By a linear subdivision of S we may furthermore assume that no 1-simplex of S contains more than one pre-image of a double point of Π. We will call such a p S X an F -adapted triangulation. A crossing datum triangulation of ∈ Π at a double point is the information saying which of the two 1-simplices of S containing the pre-images of p crosses over the other. Definition 1.4. A (PL-)link diagram D is a link projection Π dressed with crossing data at D each double point ∈ Π . The double points of p are called crossings of D .

3 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 3 3 L ⊂ which is regular with respect to π defines a link diagram D by the compo- A link R 3 where X → R of a PL-embedding : X is a finite disjoint union of PL-circles such F sition I = I ( X ) with the projection π , using the z -coordinate to define the crossing data at that L the crossings. Conversely we have: Each link diagram is the regular projection of a unique link class. Proposition 1.5. 2 : X → D ⊂ R Let be the PL-map defining the link diagram D . Equip Proof. with an F X -adapted triangulation . We may apply a linear subdivision to each 1-simplex of S which S F in three 1-simplices such that the middle D crosses over another 1-simplex at a crossing of peace contains the crossing. Therefore we may and will assume that the an overcrossing 1-simplex of is always accompanied on both sides with 1-simplices that do not contain a S crossing and whose other neighboring 1-simplex is not an overcrossing 1-simplex. We now L by modifying the simplicial map (while keeping S fixed) to an injective construct a link F 3 : = | S I R | → by moving, at each crossing point, the upper 1-vertex X simplicial map vertically upwards over a small distance into the upper half space and leaving the lower xy -plane, and adapting the map F on the neighboring 1-simplices (which do not one in the contain crossings and whose other neighbor is not an overcrossing 1-simplex) accordingly. L = Then ( X ) is a regular PL-link with respect to π , and L projects onto D . This shows I that there exists a PL-link projecting regularly onto . D ′ L is another PL-link which is regular with respect To show the uniqueness, suppose that π and which projects onto D to . Recall from week 36 that we need to fix a triangulation ′ ′ S of L in order to define ∆-moves, although the combinatorial equivalence class does not ′ − 1 -moves we can translate L vertically depend on this choice. Using a sequence of ∆ and ∆ over an arbitrary distance (the reader is invited visualize this procedure). Since such a ′ nor its equivalence class, we may assume that L translation changes neither the projection of ′ lies entirely above . Recall from week 36 that we can refine a given triangulation of L L − ′ 1 arbitrarily by application of ∆ and ∆ L moves. We may thus replace the triangulations ′ of L and L within the same combinatorial equivalence class by refinements which coincide ′ . Therefore it suffices to show that π is with each other after applying the projection L ′ under the assumption that the image of the triangulation S combinatorially equivalent to L ′ L under π is the same as the image of of under F . In this situation the 1-simplices of L S ′ ′ L are in natural bijection such that corresponding pairs σ,σ determine a trapezium T and σ ′ σ σ at the top and with two vertical sides, with the side at the bottom. If σ has no crossings ′ or is “passing under” at a crossing then does not contain points of L . Hence we can T σ ′ ′ of L replace, by two ∆-moves, the 1-simplex by the three other sides of T σ . We do this for σ ′ ′ ′′ L all 1-simplices which are undercrossings or non-crossing. The intermediate link σ of L π because of the vertical 1-simplices it thus obtained is in general not regular with respect to ′ σ contains, but this is of no concern. Now observe that each of the over-crossing 1-simplices ′ ′′ ′′ of , and that T is also a 1-simplex of does not contain any points of L L . We can now L σ ′′ L modify by combinatorial moves applied to the overcrossing 1-simplices as we did before for the undercrossing and non-crossing 1-simplicies. After these modification we will finally obtain the link L , finishing the proof of the uniqueness. From the work of the first knot tabulator Peter G. Tait (around 1877) the following basic question emerges, marking the beginning of knot theory: “When do two link diagrams define the same equivalence class of links?” A fundamental insight into the nature of this question was provided by Kurt Reidemeister (1926):

4 4 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 Ω2 Ω3 Ω1 Ω1 ∼ ∼ ∼ ∼ Reidemeister moves Figure 1. ′ ′ define the same link class iff D and D Two link diagrams D,D are connected Theorem 1.6. 2 planar ambient isotopies (i.e. PL-isotopies of by a sequence of modifications consisting of R applied to the diagram) and of the three types of local (i.e. taking place within a small disk Ω1 , without changing the diagram outside this disk) moves called Reidemeister moves of type or Ω3 and their inverses. The three types of Reidemeister moves are those illustrated in Ω2 Figure 1. Proof. We give a sketch of the proof. It is easy to see that two diagrams which differ by one of the 3 Reidemeister moves defined above, or by a planar ambient isotopy, define combinatorially equivalent links. The converse statement is less obvious and requires a serious analysis. Suppose that we modify a given link π by a ∆-move. We L which is regular with respect to ′ L is itself regular with respect to π . Indeed, may first of all assume that the modified link ′ and L from the proof of Proposition 1.2 it is clear that the set of directions such that both L project regularly along this direction is the complement of finitely many lines and points of 2 P ( R the real projective plane U of the direction ). Hence there exists an open neighborhood 3 π (in the real projective plane of directions in R ) such that for all directions of the projection in U the projection of L along this new direction is still regular. Then the projection of L along a direction in U D . But we have also seen that U contains is planar ambient isotopic to ′ L is regular. Therefore, if two diagrams define equivalent directions with respect to which links then they can be connected by a sequence of planar ambient isotopies and steps obtained − 1 by the projection of a ∆ or ∆ -move such that the link before and after the move is regular π . with respect to − 1 Now every ∆-move is the composition of ∆ and ∆ -moves associated with the six 2- simplices in the barycentric subdivision of the 2-simplex defining the ∆-move. Again, upon a small deformation of the direction of the projection and application of planar ambient π π . Since isotopies we may assume that the subdivided ∆-moves are all regular with respect to repeated barycentric subdivision produces a triangulation of the original 2-simplex with an arbitrarily small mesh (see week 36) we can make the participating ∆-moves small enough so as to ensure that the interior of π (∆) of each subdivision 2-simplex ∆ intersects nontrivially with at most one projected 1-simplex of L , or with two projected 1-simplices of L crossing each other in an interior point of of (∆). By the regularity assumptions we see that if π the projection of a 1-simplex σ of L intersects the interior of π (∆) nontrivially then either π ( σ ) intersects the boundary of π (∆) at two interior points of the faces of π ( ∂ (∆)), or σ is the neighboring 1-simplex in L of a face of ∂ (∆). Listing all possibilities for the projected ∆-moves of this kind, we finally need to establish that these modifications of D can all be realized by an appropriate combination of Reidemeister moves Ω1, Ω2 and Ω3 in combination

5 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 5 Rc Rc ∼ ∼ Diagram isotopies near a crossing Figure 2. to planar ambient isotopies of the diagram. This is not difficult; we refer to [7] for further details. Exercise (a). Let D k components. Show that one can change the be a link diagram with ′ D represents the D k - crossing data of in such a way that the resulting new link diagram component unlink. Diagram isotopies. 1.0.1. Reidemeister’s Theorem gives no description how to generate pla- ) combinatorially (by nar ambient PL-isotopies of link diagrams (called diagram isotopies 3 ). For most R “local moves”, like the role of ∆-moves for ambient isotopies of PL-links in applications this is not a problem because the invariance under diagram isotopies is otherwise easy to analyze, but for some of the later developments we will need to address this issue carefully. We will consider this problem now, while we are at it, before we return to some basic applications of Reidemeister’s result. − 1 ∆ Lemma 1.7. - and -moves applied to non- ∆ Diagram isotopies are generated by planar -simplices (these are sometimes called R0-moves), together with the planar ∆ - and 1 crossing − 1 ∆ 1 -simplex involved in a crossing (and leaving the other 1 -simplex -moves applied to a 2 involved in the same crossing fixed), provided that the -simplex defining the move does not have any other intersection with the diagram outside the two 1 -simplices involved in the cross- ing (an example of such a move is indicated in Figure 2). We call moves of the latter kind Rc -moves. Proof. The R0- and Rc-moves define diagram isotopies, as one easily sees. To see that these indeed generate all diagram isotopies we mimic the proof of Proposition [4, Proposition 1.10] 3 (also see week 36) of the corresponding result for PL-links in . This proof shows that it R suffices to prove that these local combinatorial moves generate the translations of diagrams 2 in . This is easy to see (again following [4, Proposition 1.10]). R ′ ′ and D be link diagrams, and let Lemma 1.8. L Let L D be PL-links projecting regularly , ′ onto D and D respectively via the projection π . The following assertions are equivalent: ′ D,D are connected by a planar ambient isotopy. (i) ′ − 1 (ii) are connected by a sequence of ∆ and ∆ -moves each of which has the property L,L of representing an ambient isotopy t → L π such that L is regular with respect to t t for all t ∈ [0 , 1] ). Proof. (i) implies (ii) by the above Lemma, since the R0- and Rc-moves in the Lemma are − 1 indeed projections of ∆- and ∆ -moves with the required property.

6 6 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 Figure 3. For the converse implication we proceed as in the proof of Reidemeister’s Theorem, by π . By assumption the ∆-move is the end result of projecting a sufficiently small ∆-move via a family of such moves with the property that all intermediate steps project regularly with respect to π . It is an easy verification (as in the proof of Reidemeister’s Theorem in [7]) that R such projected ∆-move (after a suitable linear subdivision if necessary) are of the kind 0 or Rc . 1.0.2. Diagram isotopies and generic diagrams. It is an extremely useful idea to think of a link diagram as being composed of horizontal strips such that each strip contains at most one crossing or local maximum or minimum, see Figure 3. To formalize this, fix a nonzero 2 the height function. Unless specified otherwise we h linear functional → R . We call h R : will choose h = y . Definition 1.9. A link diagram D is called generic (with respect to h ) if the restriction of h to Π (the link projection underlying D ) has finitely many local extrema, and if the local D extrema and the crossings all have distinct heights. We call the union of the finite set of local

7 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 7 0 R 0 R ∼ ∼ Rc Rc ∼ ∼ Figure 4. The Reidemeister moves R0 and Rc (+refl. in vert. line). ′ ′ Rc Rc ∼ ∼ Figure 5. Nongeneric diagram isotopies (+refl. in hor. line) extrema of h on a generic diagram D with the finite set of crossings of D the set of singular points (with respect to h ). D of Every link diagram is ambient isotopic to a generic link diagram. For example, for any given D , its rotations ]. link diagram will be generic for all but finitely many angles φ ∈ [0 , 2 π D φ Every generic link diagram can be cut into horizontal strips each of which contains at most one singular point, besides a number of disjoint polygonal segments without local extrema

8 8 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 with one boundary point on the bottom of the strip and its other boundary point on the top of the strip. Such strips are called elementary strips . 1] D and D Definition 1.10. be generic link diagrams. A diagram isotopy Let , D 3 t → [0 0 t 1 D to from D . is called generic if D 1] is generic for all t ∈ [0 , t 1 0 A general diagram isotopy between two generic link diagrams can be broken up as a com- position of a sequence of diagram isotopies consisting of a generic diagram isotopies in an elementary strip, or non-generic diagram isotopies of the type R0 or Rc. A move of type R0 or Rc can be non-generic: the simplest way in which this may happen involves the creation or annihilation of two consecutive elementary strips containing a local maximum and a local minimum (the R0-move) or two such elementary strips bordering an elementary strip with a crossing as shown in Figure 4 (together with the reflection of the Rc-move in a vertical line (or the plane of the diagram)). Another type of non-generic move induced by R0 and Rc-moves amounts to interchanging the relative heights of two singular points. Consider a (non-elementary) strip containing two singular points which have the same height. By diagram isotopies it can be deformed and decomposed as a composition of two elementary strips in two different ways (with a different order of the heights of the two singular points). The resulting diagram isotopy between these two compositions of elementary strips is referred to as “changing the relative heights of non- interacting singular points”. Such non-generic diagram isotopies are consequence of R0- and Rc-moves (in the sense of Figure 2) as well. Exercise (b). Consider the composition of an elementary strip with a local maximum and a strip without singular points. Show how to change the order of the two strips using R0-moves (in the sense of 2). Do the same for an elementary strip containing a crossing, using Rc- moves. Prove that this implies that “changing the relative heights of non-interacting singular points” is a consequence of R0- and Rc-moves. Conversely we have: Theorem 1.11. Any diagram isotopy between generic link diagrams is a composition of elementary moves of the following kind: (i) Generic diagram isotopies in elementary strips. (ii) Inserting or deleting a strip without singular points. (iii) Changing the relative heights of non-interacting singular points. (iv) Nongeneric R0-moves (as in Figure 4). Nongeneric Rc-moves (as in Figure 4, and its reflection in a vertical line). (v) Proof. It is easy to see that every R0- or Rc-moves is a composition of these five types of moves; conversely (as we argued in the text above) these moves are consequences of the R0- and Rc-moves. By Lemma 1.7 the result follows. Exercise (c). Prove that in Theorem 1.11 the nongeneric Rc-moves can be replaced by the nongeneric moves of type Rc’ of Figure 5. As a corollary we obtain the following refined version of Reidemeister’s Theorem: Two generic link diagrams represent isotopic links if and only if they can be Theorem 1.12. connected by a sequence of moves of the type listed in Theorem 1.11 together with the three Reidemeister moves (as listed in Figure 1). , Ω2 and Ω3 Ω1

9 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 9 1.1. Variations Two additional structures play a crucial role in various aspects of the theory. oriented link is a link in which all components are dressed with Oriented links. An 1.1.1. an orientation. Two oriented links are called equivalent if they are ambient isotopic and the isotopy preserves the orientation. It is clear that there is a corresponding notion of equivalence of oriented link diagrams. Generally this complicates the application of Reidemeister’s Theorem a little bit because one needs to consider the Reidemeister moves with all the possible orientations of the strands involved. Framed links and regular isotopy. A framed link is a smooth link L dressed with a 1.1.2. f . The framing of L determined by f is the homotopy class non-vanishing normal vector field f within the space of non-vanishing normal vector fields. An isotopy of framed links is a of smooth ambient isotopy of the underlying unframed links such that the framings correspond under the isotopy. Although this notion is most naturally defined in the smooth context, the notion can be expressed in terms of combinatorial equivalence of link diagrams as well. This leads to the regular isotopy and this will be discussed later, in relation to ribbon links . notion of First applications of Reidemeister’s Theorem 2. One of the basic goals of knot theory is to describe the set of L of equivalence classes of tame links. Reidemeister’s Theorem provides an answer by describing the set L as the quotient 2 D of planar equivalence classes of link diagrams in R of the set by the equivalence relation generated by the 3 Reidemeister moves Ω1, Ω2 and Ω3. The problem with this description lies in the long and complicated chains of Reidemeister moves and their inverses that are in general necessary to establish the equivalence of two given diagrams. Understanding the complexity of algorithms deciding on the equivalence of given diagrams (e.g. the unknotting problem) is a field of current research. The following theorem is an equivalent “dual formulation” of Reidemeister’s Theorem which is often useful in practice: A link invariant is a map Definition 2.1. : L→ V to a set V . f Theorem 2.2. A map f : D → V defines a link invariant iff f is invariant for the 3 Reidemeister moves. This is an obvious consequence of Theorem 1.6. Proof. Knowing all link invariants describes the set L . However, the above theorem is not suitable to find link invariants, but rather to verify if a given function defined on D is a link invariant. Later in the course we will discuss a sophisticated modern construction of link invariants based on the representation theory of quantum groups, and (a version of) the above theorem will indeed be used to prove the result. As such Reidemeister’s result is of great theoretical use. In this section we will discuss some classical link invariants, using Reidemeister’s Theorem when necessary.

10 10 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 The number of components. One of the first interesting link invariants is the number 2.0.3. of components of a link. It is trivial that this is an invariant of tame links (in the topolog- ical category). It is also easy to see directly that this defines an invariant of PL-links for combinatorial equivalence. The linking number is not an ordinary link invariant, since it is 2.0.4. The linking number. only defined naturally on oriented two component links. are disjoint, oriented links and suppose that K K L is regular with L Suppose that and ∪ xy -plane. Let D respect to the standard projection and D π be the link diagrams on the K L L respectively onto the xy and π , and obtained from the regular projection of K -plane via let K,L ) be the set of crossings of D V and a strand K which involve both a strand from ( L K ∪ . We define from L ∑ D 2 ,D ) ) := 1 / lk ( (2.1) c ( L K K,L ) ( V ∈ c where ( c ) = 1 if the upper strand points to the right when we pass the crossing following the lower strand, and ( c ) = − 1 otherwise. The number lk D Theorem 2.3. ,D ( ) only depends on the ambient isotopy class of the L K K L and its decomposition as a disjoint union of oriented link and L . Therefore we ∪ K Lk ( K,L ) (the linking number of K and L ). Moreover, we have may denote this number as ( K,L ) ∈ Z . Lk First of all observe that the number ( K,L ) depends only on the diagram isotopy Proof. lk D ) ,D D . Hence to establish the result it suffices to show that lk ( class of the diagram K L L K ∪ is invariant under all the (oriented versions of) the Reidemeister moves Ω1, Ω2 and Ω3. The invariance under moves of type Ω1 is obvious since the crossings involved in such moves are V ( K,L ). We leave it to the reader to perform the easy task of showing invariance never in under Ω2 and Ω3. Finally we will prove that Lk ( K,L ) ∈ Z , using that this number is invariant for ambient isotopy. By Exercise a D we may change the crossing data of ∪ K L ′ ′ ′ ∪ L projecting D represents an unlink K in such a way that the resulting link diagram ′ . But the unlink can also be represented by a link diagram without crossings. D regularly onto ′ ′ ′ ′ ( ) = lk ( D Since Lk ,D ) depends only on the ambient isotopy class, this implies that ,L K L K ′ ′ K ,L Lk ) = 0. But on the other hand, a change of a crossing datum shifts the number lk ( Lk K,L ) ∈ Z . by 1. The conclusion of these two facts is that ( The space of colorings modulo p . Let D 2.0.5. be a link diagram, and X an abelian group. L C L,X ) which will only depend on the link class L defined by We will define an abelian group ( D up to isomorphism. In the special case X = F , the additive group of the finite field with p L p elements (where p is a prime) we will give an explicit presentation for C ( L,p ) := C ( L, F ), p the “space of colorings of modulo p .” L arc after cutting the lower strands D of is defined to be a connected component of D An L L D be the set of arcs, and let . Let A at each crossing of C be the set of crossings of D . L L A coloring of D the relation is a function f : A → X such that at each crossing c ∈ C L l f ( u ) = f ( l , ) + f ( 2 c ) holds, where u denotes the arc of the upper strand at the crossing + − and l ,l denote the arcs of the two pieces of the cutted lower strand at the crossing. − +

11 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 11 [ C ] and Z [ A ] on the finite sets C and A respectively. We Consider the free abelian groups Z Z : [ C ] → Z [ A ] by define a homomorphism of abelian groups M L ) = 2 ( c (2.2) u − l M − l , + − L u,l and l ]. Then is as above) and extending this by linearly to Z [ C (where the meaning of − + -valued colorings of L,X ) of X the above description of the abelian group D ( amounts to C L C ( L,X ) := Hom ,X (Coker( M ) ) (2.3) L Z C Theorem 2.4. ) only depends on the ambient isotopy ( L,X Up to isomorphisms, the group . class of links represented by D L M Proof. ) only depends on the It is clearly sufficient to show that the abelian group Coker( L L ] of links represented by D . The presentation ambient isotopy class [ L M L C ] → (2.4) −→ Z [ A ] Z Coker( M [ ) → 0 L is clearly independent for diagram isotopies. Hence it remains to check the invariance (up to isomorphisms) for the Reidemeister moves Ω1, Ω2 and Ω3. The reader is invited to verify the details. = F , the finite field of order In the special case p (with p a prime number), it follows X p that T L, L,p ) := C ( ( F (2.5) ) = Ker( C M ) p L,p T M where denotes the transpose of the reduction modulo p of the matrix M , reducing L L,p ( L,p ) to a linear algebra problem over the field F C . the problem of computing p ∼ k ( L, 2) denotes the number of components of −−→ F Exercise (d). Show that where k C L . 2 Exercise (e). C ( L,p ) for various p , prove that Using the link invariants The 3-component borromean rings link is not equivalent to the 3-component unlink. (i) The trefoil knot and the eight knot are nontrivial and distinct. (ii) The braid group 3. The algebraic braid group. 3.0.6. alg B The algebraic braid group Definition 3.1. n strands is the group generated by n − 1 on n generators σ , and relations , i = 1 ,...,n − 1 i (i) , and σ 1 = σ > σ | σ | i − j if i j j i (ii) σ . 2 σ − σ ,...,n = 1 = σ i σ for all σ i +1 i +1 i i +1 i i alg alg S B is a quotient of Corollary 3.2. The symmetric group π sending σ by the map to n n i the transposition ( i,i + 1) .

12 12 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 The topological braid group. 3.0.7. n -strands is a disjoint union β of n segments (i.e. Definition 3.3. A topological braid on 2 3 , embeddings of × [0 , 1] ⊂ R R such that: 1] [0 ) in t ∩{ = t }| = n for all β ∈ (0 , 1) . | (i) z ∩{ z = t } (ii) { 1 ,...,n }×{ 0 }×{ t } for all t ∈{ 0 , 1 } . β = Similarly we define PL-braids and smooth braids, where for smooth braids Definition 3.4. 0 and for t ∈ [0 , ) we require that there exists an such that the strands are all vertical for > (1 − , 1] . ∈ t Definition 3.5. β , if there and β β strongly isotopic , notation β ∼ We call two braids 1 2 2 1 s β β which respects the planes z = h for all exists an ambient isotopy transforming in 2 1 t (0 1) and all t ∈ [0 , 1] , and which is the identity for all , ∈ [0 , 1] in the planes z = 0 and h ∈ = 1 . z The proof of the following result is standard and reminiscent of the proof that the funda- mental group of a topological space (with base point) is a group: The set of strong isotopy classes of topological braids on -strands form a n Theorem 3.6. top is defined by the strong isotopy class β such that the composition [ β of the ][ β ] ] [ B group n 2 1 β β braid -coordinate) by on top of obtained by putting z , and reducing the heights (the β 2 1 a factor 2 . The unit element is the braid consisting of n vertical straight segments of length one. Remark 3.7. Observe that the composition of smooth braids is well defined in this way, since the strands of the smooth braids are required to be vertical near their end points. Definition 3.8. -plane, A braid diagram is a regular projection of a PL-braid onto the xz y -axes. where regularity refers to the projection π alon the y The following crucial result is nontrivial. [1] ) Two braid diagrams D D and Theorem 3.9. D are strongly isotopic iff (See and D 1 2 1 2 are connected by a finite sequence of moves consisting of braid diagram isotopies (i.e. ambient z diagram isotopies respecting the height function Ω2 and Ω3 . ) and the Reidemeister moves The map Corollary 3.10. : σ ) extends to an isomorphism → X 1 (for i = 1 ,...,n − t i i alg top : -strands where all but two strands are vertical → B X n . Here t B is the braid on n n i segments, except for the strands i and i + 1 which cross each other (and the strand from ( i, 0 , 0) to ( i + 1 , 0 , 1) crosses over the other strand). top Proof. are generators of the group B X The . Indeed, we can represent any braid β on n i -strands such that all crossings have distinct heights, and cutting such a braid in horizontal n strips such that every strip contains at most one crossing gives a representation of β as a ± 1 ) product of elements of the form ( X . The generators X satisfy the Reidemeister equiva- i i alg top t : B → . The injectivity of lence Ω B , and thus we obtain a surjective homomorphism n n 3 the map follows from Artin’s Theorem 3.9. top π : B We define a map )( i by requiring that π ( β → S ) = j if the strand of β connected n n top i, 0 , 0) ends in ( j, 0 , 1). It is easy to see that π to the point ( is a surjective homomorphism. alg top ◦ = π It follows easily that π t by checking it on the generators σ . i

13 QUANTUM GROUPS AND KNOT THEORY: WEEK 37 13 top top top The kernel Definition 3.11. ⊂ of π B is called the pure braid group. P n n There exists a canonical short exact sequence Corollary 3.12. top π top top S P 1 (3.1) B −→ → → → 1 n n n n . Let Y = 3.0.8. C The braid group and configuration spaces of points in \∪ C { z = i i