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This paper refines previous work by the first author. We study the questionof which links in the 3-sphere can be obtained as closures of a given1-manifold in an unknotted solid torus in the 3-sphere (or genus-1 tangle) byadjoining another 1-manifold in the complementary solid torus. We distinguishbetween even and odd closures, and define even and odd versions of the Kauffmanbracket ideal. These even and odd Kauffman bracket ideals are used to obstructeven and odd tangle closures. Using a basis of Habiro's for the even Kauffmanbracket skein module of the solid torus, we define bases for the even and oddskein module of the solid torus relative to two points. These even and oddbases allow us to compute a finite list of generators for the even and oddKauffman bracket ideals of a genus-1 tangle. We do this explicitly for threeexamples. Furthermore, we use the even and odd Kauffman bracket ideals toconclude in some cases that the determinants of all even/odd closures of agenus-1 tangle possess a certain divisibility.

Even and odd Kauffman bracket ideals for genus-1 tangles