Skein theory for SU(n)-quantum invariants

For any n ≥ 2 we define an isotopy invariant, 〈Γ〉n , for a certain set of n-valent ribbon graphs Γ in R, 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 3manifold M and we prove that it determines the SLn -character variety of π1(M). AMS Classification 57M27; 17B37