Skein theory for SU(n)-quantum invariants

Details Skein theory for SU(n)-quantum invariants Skein theory for SU(n)-quantum invariants

2004-07-16

arxiv.org/abs/math/0407299v4

Algebr. Geom. Topol. 5 (2005) 865-897

Mathematics

Quantum Algebra

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-36.abs

Scientific paper

For any n>1 we define an isotopy invariant, _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 SU_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 <.>_n, we define the SU_n-skein module of any 3-manifold M and we prove that it determines the SL_n-character variety of pi_1(M).

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