  Given a compact oriented 3-manifold M in S^3 with boundary, an (M,2n)-tangle
T is a 1-manifold with 2n boundary components properly embedded in M. We say
that T embeds in a link L in S^3 if T can be completed to L by a 1-manifold
with 2n boundary components exterior to M. The link L is called a closure of T.
We define the Kauffman bracket ideal of T to be the ideal I_T generated by the
reduced Kauffman bracket polynomials of all closures of T. If this ideal is
non-trivial, then T does not embed in the unknot. We give an algorithm for
computing a finite list of generators for the Kauffman bracket ideal of any
(S^1 x D^2, 2)-tangle, also called a genus-1 tangle, and give an example of a
genus-1 tangle with non-trivial Kauffman bracket ideal. Furthermore, we show
that if a single-component genus-1 tangle S can be obtained as the partial
closure of a (B^3, 4)-tangle T, then I_T = I_S.
