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

Abstract

For any $n\ge 2$ we define an isotopy invariant, ${\langle \Gamma \rangle }_n,$ 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 ${\langle \cdot \rangle }_n,$ 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 ${\pi }_1\left(M\right).$