8.2 Skein relations

We shall now show formally how a 3-manifold Y with dY = X gives rise to a vector

 

As mentioned in Chapter 1 the Jones polynomial of links in S3 can be characterized by a skein relation. Thisinvolves comparison of the three links obtained by various crossing changes at a fixed vertex of a planar diagram. The identification of Witten's functional integral invariant (for G = SU(2) with its standard representation C2) with a value of the Jones polynomial (t = 2iri/(k + 2)) rests therefore on demonstrating that it satisfies the same skein relation.

For large k we always get the whole space. In particular take r = 4 and A, = A2 = A* = A*. Then the dimension of the

The fact that Witten's invariant satisfies a skein relation of the right form (for SU(n) with its standard representation C") is an elementary consequence of the fact that the Hilbert space of Witten's theory for the 2-sphere S2 with four marked points (two positive, two negative) has dimension 2. In fact decomposing S3 into two balls by cutting out a small neigh­bourhood of the given vertex we get precisely S2 with four marked points as common boundary. In its Hilbert space "M we have a vector, say u, determined by the exterior, and three vectors, say v+, v0, determined by the three interiors (depending on the links L+, L_, L0). The Witten invariants for these three links are then the scalar products in

 

 

If dim "X = 2 the three vectors v+, , v0 must satisfy a linear relation and their scalar products with u then satisfy the same relation. Note that X and the three vectors v+, v , v0 are locally determined and are independent of the rest of the link. Thus the coefficients of the linear relation are universal (depending only on n and k).

The reason why dim X = 2 is the following. Quite generally the Hilbert space for S2 with points Pf (i= 1,..., r) marked by representations Af of G can (from its definition) be shown to be a subspace of the G-invariant part of the tensor product

 

 

G-invariant part of

 

 

is the number of irreducible summands in A!® A^ For G = SU(n) and A, = C" this number is 2.

The computation of coefficients of the skein relation (i.e. the dependence of n and k) is given by Witten [36]. It depends on algebraic results of Verlinde [34] which Witten reinterprets in terms of surgery formulae. These ideas will be discussed briefly in the next section.