#### Algebraic and Geometric Topology 5 (2005),
paper no. 36, pages 865-897.

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

### Adam S. Sikora

**Abstract**.
For any n>1 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 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).
**Keywords**.
Kauffman bracket, Kuperberg bracket, Murakami-Ohtsuki-Yamada bracket, quantum invariant, skein module

**AMS subject classification**.
Primary: 57M27.
Secondary: 17B37.

**E-print:** `arXiv:math.QA/0407299`

**DOI:** 10.2140/agt.2005.5.865

Submitted: 23 July 2004.
Accepted: 9 May 2005.
Published: 29 July 2005.

Notes on file formats
Adam S. Sikora

Department of Mathematics, University at Buffalo

Buffalo, NY 14260-2900, USA

Email: asikora@buffalo.edu

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