Christian Blanchet

Three-manifold invariants derived from the Kauffman bracket

IN SEVERAL recent articles, Lickorish [ 11, 12,133 has given an elementary construction of 3manifold invariants using the one variable Kauffman bracket [6] evaluated at 4r-th roots of unity, I 2 3. We show that evaluations of the bracket at 2p-th roots of unity, p odd, also give 3-manifold invariants. Moreover, we show that no other evaluations at other values lead to invariants. In [13], Lickorish has described the precise relationship of the invariants given by his formulas, to the Jones-Witten SU(2), invariants [18] as established by Reshetikhin and Turaev [14] using quantum groups, and further studied by Kirby and Melvin [8, 91. In the spirit of the present paper, Blanchet has given a construction of invariants for 3dimensional Spin manifolds. He obtains invariants for evaluations at all primitive k-th roots of unity for k f 8 mod 16 [2] (compare [9], [16]). The construction of the invariants is based on Kirby’s theorem [7] describing how two surgery descriptions of the same oriented closed 3-manifold are related. If a manifold M3 is represented by a banded link L in the 3-sphere, then the invariants are given by finite linear combinations of Kauffman brackets of tablings of L, to be evaluated at primitive 2p-th roots of unity. In fact, our invariant for odd p can be expressed by a generalization of the formula given by Lickorish in [13]. However, we do not need to use the Temperley-Lieb algebra, which was Lickorish’s main tool in establishing the existence of his invariants. Instead we systematically study which tablings and which evaluations of the Kauffman brackets of these tablings are invariant under Kirby’s calculus. This enables us to show first that non-trivial invariants can only exist for evaluations at primitive 2p-th roots of unity, and second that they do exist and are essentially unique for each such evaluation. Here is an outline of this paper. We define the Jones-Kauffman module K(M) of an oriented 3-manifold to be the Z [ A, A ‘]-module generated by banded links in M, divided by the usual Kauffman relations (see [4] for an overview of more general skein modules). Let g denote the Jones-Kauffman module of the solid torus. (In [ 131, essentially the same object, after a change of coefficients from Z [ A, A -‘I to C, is called 2l.) Given a ncomponent banded link in S 3, there is a n-linear form (, . . . ,)L on 9’” given by replacing the components of L by elements of a, and taking the bracket of the resulting banded link in S3. We call this n-linear form the meta-bracket of L. (It corresponds to the map (IQ, in [13].) In particular, we have a symmetric bilinear form ( , ) on a given by the meta-bracket of the banded Hopf link where each component has writhe zero. Let t be the self-map of .!Z# induced by one positive twist. The following observation is due to Lickorish and appeared in [13] with a slightly different normalization. Suppose we can find an element R l g with

Hecke algebras, modular categories and 3-manifolds quantum invariants

Abstract We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin–Turaev invariants of closed 3-manifolds constructed from the quantum groups Uqsl(N) by Reshetikhin–Turaev and Turaev–Wenzl, and from skein theory by Yokota. The possibility of such a construction was suggested by Turaev, as a consequence of Schur–Weil duality. We then discuss the choice of the framing parameter. This leads, for any rank N and level K, to a modular category H N,K and a reduced invariant τ N,K . If N and K are coprime, then this invariant coincides with the known invariant τPSU(N) at level K. If gcd (N, K)=d>1 , then we show that the reduced invariant admits spin or cohomological refinements, with a nice decomposition formula which extends a theorem of H. Murakami.

Topological quantum field theories for surfaces with spin structure

Reened quantum invariants for closed three-manifolds with links and spin structures are extended to a Topological Quantum Field Theory. By a `universal con-struction, one associates, to surfaces with structure, modules which are shown to be free of nite rank. These modules satisfy the multiplicativity axiom of TQFT in an extended Z=2-graded sense, and their ranks are given by a spin reened version of thèVerlinde formula. The relationship with thèunspun theory is given by a naturaìtransfer map.

Modular categories of types B,C and D

Abstract. We construct four series of modular categories from the two-variable Kauffman polynomial, without use of the representation theory of quantum groups at roots of unity. The specializations of this polynomial corresponding to quantum groups of types B, C and D produce series of pre-modular categories. One of them turns out to be modular and three others satisfy Bruguières modularization criterion. For these four series we compute the Verlinde formulas, and discuss spin and cohomological refinements.

Refined quantum invariants for three-manifolds with structure

Introduction. Following Witten’s interpretation ([Wi]) of the Jones polynomial ([Jo]) in terms of Topological Quantum Field Theory , Reshetikhin and Turaev ([RT]) and then many others have constructed invariants of 3-manifolds now called Quantum Invariants (see [Tu2] for a detailed exposition, and [Vo] for a survey). The construction of Reshetikhin and Turaev involves representation theory of quantum groups. This point of view gives a deep insight into the algebraic questions related to the subject, however it is not immediately accessible for the beginner. Among these quantum invariants those called the SU(2)-invariants can be obtained easily from the skein theory associated with the Kauffman bracket ([Ka]). This was first observed by Lickorish ([Li1],[Li2],[Li3]) and then systematically studied in [BHMV1]. Section 1 deals with this skein method. Starting with a formal skein theory, we discuss the construction of 3-manifolds invariants, and give the simplest examples. We think that this could be helpful for the beginner and hope that the method will be applied to new examples. Once one has constructed a lot of 3-manifold invariants, the question is to understand their meaning, and this is far from clear at the moment. Let us discuss the example of τ at q=e iπ 8 ([KM]) which corresponds to θ8 in [BHMV1] and [Bl1]. This invariant decomposes as a sum, over all spin structures on the manifold, of spin invariants. Moreover the spin invariant is (a version of) the well known Rochlin invariant. This was first observed by Kirby and Melvin and generalized independently in [KM], [Tu1] and [Bl1]. This example shows that considering refined invariants can help in understanding their geometrical meaning. Section 2 is about cohomological refinements of quantum invariants. According to H. Murakami ([Mu]) τ SU(n) r admits such refinements, for conveniently chosen