Commentarii Mathematici Helvetici

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Volume 72, Issue 4, 1997, pp. 521–542
DOI: 10.1007/s000140050032

Published online: 1997-12-31

Rings of SL2( $\Bbb{C}$ )-characters and the Kauffman bracket skein module

D. Bullock[1]

(1) Boise State University, USA

Let M be a compact orientable 3-manifold. The set of characters of SL2($\Bbb {C}$)-representations of $ \pi_1(M) $ forms a closed affine algebraic set. We show that its coordinate ring is isomorphic to a specialization of the Kauffman bracket skein module, modulo its nilradical. This is accomplished by realizing the module as a combinatorial analog of the ring in which tools of skein theory are exploited to illuminate relations among characters. We conclude with an application, proving that a small manifold's specialized module is necessarily finite dimensional.

Keywords: Knot, link, skein theory, representation theory, 3-manifold

Bullock D.: Rings of SL2( $\Bbb{C}$ )-characters and the Kauffman bracket skein module. Comment. Math. Helv. 72 (1997), 521-542. doi: 10.1007/s000140050032