The Kauffman skein module at first order

TL;DR

This paper explores the first-order behavior of the Kauffman skein module for 3-manifolds with boundary, proposing a conjecture linking it to Reidemeister torsion and introducing a twisted self-linking concept.

Contribution

It introduces a conjectural relation between the first-order Kauffman module and Reidemeister torsion, and develops a twisted self-linking notion satisfying Kauffman relations.

Findings

Proved the conjecture in particular cases.

Defined and analyzed twisted self-linking satisfying Kauffman relations.

Connected skein modules to Reidemeister torsion in the first-order setting.

Abstract

For a 3-manifold $M$ with boundary, we study the Kauffman module with indeterminate equal to $-1+\epsilon$ where $\epsilon^2=0$. We conjecture an explicit relation between this module and the Reidemeister torsion of $M$ which we prove in particular cases. As a maybe useful tool, we then introduce a notion of twisted self-linking and prove that it satisfies the Kauffman relations at first order. These questions come from considerations on asymptotics of quantum invariants.