On the character rings of twist knots
Bull. Korean Math. Soc. 2011 Vol. 48, No. 3, 469-474
Published online May 1, 2011
Fumikazu Nagasato
Meijo University
Abstract : The Kauffman bracket skein module $\mathcal K_t(M)$ 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 ${\rm SL}_2(\mathbb C)$-character ring of the fundamental group of $M$. Our proof is based on some properties of Chebyshev polynomials.
Keywords : character variety, character ring, Chebyshev polynomial, Kauffman bracket skein module
MSC numbers : Primary 57M27; Secondary 57M25
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