Let Gn be the fundamental group of the exterior of the rational link C(2n) in Conway's normal form, see [7]. A presentation for Gn is given by \langle a, b|(ab)n = (ba)n\rangle [3, Thm. 2.2]. We study the character variety in SL(2, C) of the group Gn. In particular, we give the defining polynomial of the character variety of Gn. As an application, we show a well-known result that Gn and Gm are isomorphic only when n = m. Also as a consequence of the main theorem of this paper, we give a basis of the Kauffman bracket skein module of the exterior of the rational link C(2n) modulo its (A + 1)-torsion.

Let Gn be the fundamental group of the exterior of the rational link C(2n) in Conway's normal form, see [7]. A presentation for Gn is given by \langle a, b|(ab)n = (ba)n\rangle [3, Thm. 2.2]. We study the character variety in SL(2, C) of the group Gn. In particular, we give the defining polynomial of the character variety of Gn. As an application, we show a well-known result that Gn and Gm are isomorphic only when n = m. Also as a consequence of the main theorem of this paper, we give a basis of the Kauffman bracket skein module of the exterior of the rational link C(2n) modulo its (A + 1)-torsion.