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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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