Skein theory for SU(n)–quantum invariants

For any $n \geq 2$ we define an isotopy invariant, ${〈 Γ 〉}_{n},$ for a certain set of $n$ –valent ribbon graphs $Γ$ 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 ${〈 \cdot 〉}_{n},$ 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 $π_{1} (M) .$

Keywords

Kauffman bracket, Kuperberg bracket, Murakami–Ohtsuki–Yamada bracket, quantum invariant, skein module

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 17B37

References

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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

Volume 5, issue 3 (2005)

Adam S Sikora

Abstract