Generalizing the relation between the Kauffman bracket and Jones polynomial

Abstract

We generalize Kauffman’s famous formula defining the Jones polynomial of an oriented link in [Formula: see text]-space from his bracket and the writhe of an oriented diagram [L. Kauffman, State models and the Jones polynomial, Topology 26(3) (1987) 395–407]. Our generalization is an epimorphism between skein modules of tangles in compact connected oriented [Formula: see text]-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 [Formula: see text]-analog of the first homology group of a [Formula: see text]-manifold [J. Przytycki, A [Formula: see text]-analogue of the first homology group of a [Formula: see text]-manifold, in Contemporary Mathematics, Vol. 214 (American Mathematical Society, 1998), pp. 135–144]. In certain cases, e.g., for links in submanifolds of rational homology [Formula: see text]-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.