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On the character rings of twist knots
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The Kauffman bracket skein module K_t(M) of a 3-manifold M becomes an algebra for t=-1. We prove that this algebra has no non-trivial nilpotent elements for M being the exterior of the twist knot in 3-sphere and, therefore, it is isomorphic to the SL_2(C)-character ring of the fundamental group of M. Our proof is based on some properties of Chebyshev polynomials.
Abstract
The Kauffman bracket skein module K_t(M) of a 3-manifold M becomes an algebra for t=-1. We prove that this algebra has no non-trivial nilpotent elements for M being the exterior of the twist knot in 3-sphere and, therefore, it is isomorphic to the SL_2(C)-character ring of the fundamental group of M. Our proof is based on some properties of Chebyshev polynomials.
참고문헌 (7)
- M. Abramowitz, "Handbook of Mathematical Functions with Formulas" (1972), Graphs, and Mathematical Tables
- D. Bullock, "Rings of SL2(C)-characters and the Kauffman bracket skein module", Com-ment. Math. Helv, 72(4), pp.521~542 (1997)
- D. Bullock, "The Kauffman bracket skein module of a twist knot exterior", Algebr. Geom. Topol, 5, pp.107~118 (2005)
- M. Culler, "Varieties of group representations and splittings of 3-manifolds", Ann. of Math. (2), 177(1), pp.109~146 (1983)
- R. Gelca, "Some results about the Kauffman bracket skein module of the twist knot exterior", J. Knot Theory Ramiffications, 15(8), pp.1095~1106 (2006)
- J. Przytycki, "Skein modules of 3-manifolds", Bull. Pol. Acad. Sci. Math, 39(1), pp.91~100 (1991)
- J. H. Przytycki, "On skein algebras and Sl2(C)-character varieties", Topol-ogy, 39(1), pp.115~148 (2000)
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