The skein module of torus knots complements

We compute the Kauffman skein module of the complement of torus knots in S3. Precisely, we show that these modules are isomorphic to the algebra of Sl(2, C)-characters tensored with the ring of Laurent polynomials. Skein modules were introduced indenpendantly by V. Turaev in 1988 and J. Przytycki in 1991 (see [TU88, HP92]) as a C[A±1]-module associated to a 3-manifold M generated by banded links inside M with local relations known as Kauffman relations, see for instance [HP92]. In the case of M = S this construction reduces to the Jones polynomial and in the general case, the evaluation of the skein module at roots of unity is known to fit with the Topological Quantum Field Theory constructed in [BHMV]. At the same time, skein modules were investigated for themselves. It was shown in [TU, B97, PS00] that the skein modules of thickened surfaces Σ× [0, 1] were non-commutative algebras quantizing in some sense the trace functions on the Sl(2,C)-characters of Σ. D. Bullock proved in [B97] that the skein module of M for A = −1 was a commutative algebra isomorphic up to nilpotents to the algebra of trace functions on the Sl(2,C)-characters of M . Moreover, skein modules were computed in [BL05, B94] for specific 3-manifolds as S × S, lens spaces, complement of (2, 2p + 1)-torus knots and twist knots. Except for S×S, skein modules were shown to be free over C[A±1] and not to have nilpotents when putting A = −1. Unfortunately we do not have a general criterium for deciding when this should hold. Some work of C. Frohman, R. Gelca and S. Garoufalidis [FGL02, G08] showed a relation between skein modules and recurrence relations satisfied by colored Jones polynomials. T. Q. T. Lê gave in [L06] an efficient description of the skein module of the complement of two-bridge knots. These knots include the previously known examples and the structure of their skein module were ∗Centre de Mathématiques Laurent Schwartz, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France, email: [email protected]