Even and odd Kauffman bracket ideals for genus-1 tangles

TL;DR

This paper introduces even and odd Kauffman bracket ideals for genus-1 tangles, providing new tools to obstruct and analyze the closures of tangles in the 3-sphere, with explicit computations and divisibility results.

Contribution

It defines and computes even and odd Kauffman bracket ideals for genus-1 tangles, extending skein theory to distinguish and obstruct tangle closures.

Findings

Explicit bases for even and odd skein modules were constructed.

Finite generators for the even and odd Kauffman bracket ideals were obtained.

Divisibility properties of determinants of closures were established.

Abstract

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