3.6 The quantum group link invariants.
3.6.2 A basic construction method.
LetL=L1 tL
2
ttLk be a framed link, with diagram D. The diagram can
be regularly isotopped so that it lies in levels. In each level all but two of the strings will run through parallel. One pair of strings will either cross over, form a cap or form a cup.
We colourLby assigning a representation ofUq(sl(N)),Vi, to each component
Li. At any point in the diagram where a string is running from top to bottom
of a level it is coloured with the representation colouring that component. If the string has the reverse orientation it is coloured by the dual module.
Each of these layers will determine a module homomorphism, which will be built up from the elementary tangles of crossings, cups and caps. The pictorial rules in Figure 3.2 describe how we can compose them. LetT be a coloured tan- gle, i.e. a tangle with an irreducible module assigned to each of its components. We denote by J(T) the module homomorphism obtained from the composition of the module homomorphism determined by each elementary layer. Note we can only dene J(TS) if the colourings are compatible. In Figure 3.2 we re- quire l = m and V0
i = Wi for i = 1:::m to be able to compose the module
homomorphisms in this way.
T J(T) :V1 Vk !V 0 1 V 0 l S J(S) :W1 Wm !W 0 1 W 0 n; T S J(TS) =J(T)J(S); T S J(T S) =J(T)J(S):
Figure 3.2: The pictorial composition of the module homomorphisms. Applying these to a link diagram we can construct a module homomorphism, from the scalars to the scalars, dependent on the diagram and the colouring.
3.6.3 Example.
Figure 3.3 shows a gure-eight knot, K, coloured by the moduleV and arranged into layers. Each layer determines a module homomorphism from the module at the top of the layer to the module at the bottom, as indicated on the right hand side. Taking the compositions of these, the knot K determines a module homomorphism from the scalars at the top of the picture to the scalars at the bottom.
V C[s1=N] # V V # V V V V # V V V V # V V V V # V V V V # V V V V # V V # C[s1=N]
Figure 3.3: The gure-eight knot arranged in layers.
3.6.4 Denitions.
We dene the elementary module homomorphisms for each of the elementary tangles. The denitions given here are those of [RT2]. A detailed description of the Uq(sl(2)) case can be found in [KM].
The straight string will correspond to the identity homomorphism of the colour.
V = IdV
LetRV;W be the map assigned to the positive crossing
V W where RV;W =V;WR:V W ! W V : xy 7! X tiysix 51
The negative crossing coloured byV andW (as shown below) will correspond to the homomorphismR 1
W;V .
V W
Although neitherV W nor Rare module homomorphisms, it turns out that their
composite is. The cup and cap homomorphisms are given in Figure 3.4, with the assumption that the string oriented downward is coloured byV.
:V V !C[s 1=N] f v 7!f(v) :V V !C[s 1=N] vf 7!f((uy 1):v) : C[s1=N] !V V 17! P ieiei : C[s1=N] !V V 17! P iei(yu 1)e i
Figure 3.4: The cup and cap homomorphisms.
3.6.5 Theorem.[RT1]
The maps RV W, R 1
V W, , , and are module homomorphisms. They
satisfy the identities described pictorially by the Reidemeister moves RII and
RIII.
They also satisfy the pictorial identities of Figure 3.5 (and those obtained from Figure 3.5 by changing the sign of the crossing) with all possible colourings and orientations.
The relations of Figure 3.6 also hold for all possible choices of orientation. 52
= = =
(a) (b)
= =
(c) (d)
Figure 3.5: The pictorial isotopy relations between the homomorphisms.
X V W X V W = (e) V W = V W V W = V W (f) (g)
Figure 3.6: Further relations between the module homomorphisms. LetJ(L;V1;V2;
;Vk) denote the module homomorphism determined by the
link L, coloured by V1, :::,Vk. Then J(L;V1;V2;
;Vk) is a regular isotopy in-
variant.
3.6.6 Comment.
If the tangleT below is coloured by a simple module then, by Schur's Lemma, it must represent either the zero homomorphism or a scalar multiple of the identity.
T =
The Whitney trick (shown in Figure 1.1) implies that the homomorphism deter- mined by T must be a non-zero isomorphism, hence
V = f V
for some non-zero scalar f.
3.6.7 Proposition.
LetT be a coloured (n;n) tangle with associated module homomorphism J(T). LetL be a coloured link which is the closure of T. Then
J(L) = trqJ(T):
where trq is as dened in Denition 3.2.4.
3.6.8 Proposition.
Let Li be a component of an oriented link L. Let Vi be the colour assigned to
Li. Dene L to be the link Lwith the orientation of Li reversed and the colour
Vi replaced by V i . J(L;V1;:::;Vi 1;Vi;Vi+1;:::Vk) =J(L;V1;:::;Vi 1;V i ;Vi+1;:::Vk) 54