Volume 5, issue 3 (2005)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Skein theory for $SU(n)$–quantum invariants

Adam S Sikora

Algebraic & Geometric Topology 5 (2005) 865–897

arXiv: math.QA/0407299

Abstract

For any n 2 we define an isotopy invariant, Γn, for a certain set of n–valent ribbon graphs Γ in 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 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 3–manifold M and we prove that it determines the SLn–character variety of π1(M).

Keywords
Kauffman bracket, Kuperberg bracket, Murakami–Ohtsuki–Yamada bracket, quantum invariant, skein module
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 17B37
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Publication
Received: 23 July 2004
Accepted: 9 May 2005
Published: 29 July 2005
Authors
Adam S Sikora
Department of Mathematics
University at Buffalo
Buffalo NY 14260-2900
USA