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We construct an invariant of framed links in a handlebody by means of the spin j representation of U q (sl(2, C)). We can see this invariant is an extension of the Jones polynomial and Kauffman's Dubrovnik polynomial. Moreover, we can obtain a linear representation of the generalized braid group associated with the Lie algebras of types B and C by applying the operators used in constructing an invariant to tangles in a solid torus

An invariant of links in a handlebody associated with the spin 𝑗 representation of 𝑈_{𝑞}(𝔰𝔩(2,𝐂))