Skein theory for SU(n)–quantum invariants

Sikora, Adam S¹

¹Department of Mathematics, University at Buffalo, Buffalo NY 14260-2900, USA

Algebraic and Geometric Topology, Tome 5 (2005) no. 3, pp. 865-897

Résumé

For any n ≥ 2 we define an isotopy invariant, 〈Γ〉n, for a certain set of n–valent ribbon graphs Γ in ℝ3, 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 SUn–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 SUn–skein module of any 3–manifold M and we prove that it determines the SLn–character variety of π1(M).

Détail

DOI : 10.2140/agt.2005.5.865

Keywords: Kauffman bracket, Kuperberg bracket, Murakami–Ohtsuki–Yamada bracket, quantum invariant, skein module

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