An intrinsic approach to invariants of framed links in 3-manifolds

TL;DR

This paper introduces an intrinsic method to analyze invariants of framed links in certain 3-manifolds, providing new constructions and calculations related to the Kauffman skein module and polynomial.

Contribution

It offers a novel intrinsic approach to link invariants in 3-manifolds and constructs a new version of the classical Kauffman polynomial.

Findings

Calculated the dual of the Kauffman skein module in specific 3-manifolds.

Provided a new construction of the classical Kauffman polynomial for links in S^3.

Extended the understanding of link invariants in irreducible 3-manifolds.

Abstract

We study framed links in irreducible 3-manifolds that are $Z$-homology 3-spheres or atoroidal $Q$-homology 3-spheres. We calculate the dual of the Kauffman skein module over the ring of two variable power series with complex coefficients. For links in $S^3$ we give a new construction of the classical Kauffman polynomial.