Algebraic and Geometric Topology 5 (2005), paper no. 36, pages 865-897.

Skein theory for SU(n)-quantum invariants

Adam S. Sikora

Abstract. For any n>1 we define an isotopy invariant, [Gamma]_n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the Kuperberg's bracket for n=3. Furthermore, we prove that for any n, our bracket of a link L is equal, up to normalization, to the SU_n-quantum invariant of L. We show a number of properties of our bracket extending those of the Kauffman's and Kuperberg's brackets, and we relate it to the bracket of Murakami-Ohtsuki-Yamada. Finally, on the basis of the skein relations satisfied by [.]_n, we define the SU_n-skein module of any 3-manifold M and we prove that it determines the SL_n-character variety of pi_1(M).

Keywords. Kauffman bracket, Kuperberg bracket, Murakami-Ohtsuki-Yamada bracket, quantum invariant, skein module

AMS subject classification. Primary: 57M27. Secondary: 17B37.

E-print: arXiv:math.QA/0407299

DOI: 10.2140/agt.2005.5.865

Submitted: 23 July 2004. Accepted: 9 May 2005. Published: 29 July 2005.

Notes on file formats

Adam S. Sikora
Department of Mathematics, University at Buffalo
Buffalo, NY 14260-2900, USA

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