Generalizing the relation between the Kauffman bracket and Jones polynomial

TL;DR

This paper extends the relationship between the Kauffman bracket and Jones polynomial to skein modules in 3-manifolds, introducing new modules and epimorphisms that generalize classical link invariants.

Contribution

It introduces a generalized framework connecting Jones and Kauffman bracket skein modules via epimorphisms in 3-manifolds, including graded quotients and new skein modules.

Findings

Defined epimorphisms between skein modules in 3-manifolds.

Constructed graded quotients of skein modules based on homology.

Established relations between Jones and Kauffman bracket modules in specific 3-manifolds.

Abstract

We generalize Kauffman's famous formula defining the Jones polynomial of an oriented link in 3-space from his bracket and the writhe of an oriented diagram. Our generalization is an epimorphism between skein modules of tangles in compact connected oriented 3-manifolds with markings in the boundary. Besides the usual Jones polynomial of oriented tangles we will consider graded quotients of the bracket skein module and Przytycki's q-analogue of the first homology group of a 3-manifold. In certain cases, for example for links in submanifolds of rational homology 3-spheres, we will be able to define an epimorphism from the Jones module onto the Kauffman bracket module. For the general case we define suitably graded quotients of the bracket module, which are graded by homology. The kernels define new skein modules measuring the difference between Jones and bracket skein modules. We also discuss gluing in this setting.