Bulletin of the
Korean Mathematical Society BKMS

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