# week37

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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 .

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

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 ) 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