The Kauffman bracket ideal for genus-1 tangles

Abstract

Given a compact oriented 3-manifold M in S3 with boundary, an (M, 2n)-tangle [Formula: see text] is a 1-manifold with 2n boundary components properly embedded in M. We say that [Formula: see text] embeds in a link L in S3 if [Formula: see text] can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of [Formula: see text]. We define the Kauffman bracket ideal of [Formula: see text] to be the ideal [Formula: see text] of ℤ[A, A-1] generated by the reduced Kauffman bracket polynomials of all closures of [Formula: see text]. If this ideal is non-trivial, then [Formula: see text] does not embed in the unknot. We give an algorithm for computing a finite list of generators for the Kauffman bracket ideal of any (S1 × D2, 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 [Formula: see text] can be obtained as the partial closure of a (B3, 4)-tangle [Formula: see text], then [Formula: see text].