Abstract
In [3] Hoste and Przytycki defined a two variable polynomial invariant for 1-trivial dichromatic links by a method similar to that of Kauffman in defining the Jones polynomial. In this paper we view the invariant of Hoste and Przytycki as an invariant for knots and links in the solid torus, and we give a state sum formula for this invariant. Then we define Jones’s polynomial for links in the handlebody. In particular, we give formulas for the Jones’s polynomial for knots and links in the handlebody with two handles. Moreover, we give a state sum formula for this invariant.
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