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Skein theory for \(SU(n)\)-quantum invariants. (English) Zbl 1087.57008
The paper gives a skein theoretic formulation of the Reshetikhin-Turaev quantum \(sl(n)\)-invariants for a set (the “\(n\)-webs”) of \(n\)-valent ribbon graphs \(\Gamma\) in \(\mathbb{R}^3\) including all framed oriented links. This result (Theorem 1, Proposition 2, and section 2) is proved in sections 5 and 6, after some algebraic preliminaries on the defining representation of the quantum group \(U_q(sl(n))\) and its Hecke algebra in section 4.
The resulting isotopy invariant \(\left<\Gamma\right>_n \in \mathbb{Z}[q^{1/n},q^{-1/n}]\) is shown to coincide with the Kauffman bracket of unoriented framed links for \(n=2\) and the Kuperberg bracket of \(3\)-webs for \(n=3\) (sections 1.3-1.4), and it determines the Kauffman-Vogel bracket of singular oriented framed links (section 1.5) and the Murakami-Ohtsuki-Yamada invariants of \(3\)-valent framed oriented graphs endowed with an integer flow (sections 1.7, 9 and 10).
A state sum formula for \(\left<\;\right>_n\) is given, showing that for \(n\)-web diagrams with no crossings all integer coefficients are non-negative (sections 1.6 and 8; some connections with Khovanov-Rozansky homology are mentioned). Moreover, the \(SU_n\)-skein module of any orientable \(3\)-manifold is defined, thus generalizing previous results of Frohman-Zong [preprint] and T. Ohtsuki and S.Yamada [J. Knot Theory Ramifications 6, No. 3, 373–404 (1997; Zbl 0949.57011)] for \(SU_3\). A consequence of a theorem of the author in [Trans. Am. Math. Soc. 353, No. 7, 2773–2804 (2001; Zbl 1046.57014)] is that for \(q^{1/n}=1\) and an algebraically closed field of coefficients of characteristic zero, the \(SU_n\)-skein module coincides with the coordinate ring of the \(SL_n\)-character variety of \(\pi_1(M)\).

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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