#### Volume 5, issue 3 (2005)

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Skein theory for $SU(n)$–quantum invariants

Algebraic & Geometric Topology 5 (2005) 865–897
 arXiv: math.QA/0407299
##### Abstract

For any $n\ge 2$ 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 $S{U}_{n}$–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 ${〈\cdot 〉}_{n},$ we define the $S{U}_{n}$–skein module of any $3$–manifold $M$ and we prove that it determines the $S{L}_{n}$–character variety of ${\pi }_{1}\left(M\right).$

##### Keywords
Kauffman bracket, Kuperberg bracket, Murakami–Ohtsuki–Yamada bracket, quantum invariant, skein module
Primary: 57M27
Secondary: 17B37