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

Abstract

We construct an invariant of framed links in a handlebody by means of the spin j j representation of U q ( s l ( 2 , C ) ) {U_q}(\mathfrak {s}\mathfrak {l}(2,\mathbb {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 B and C C by applying the operators used in constructing an invariant to tangles in a solid torus.