Non-semisimple quantum invariants and abelian classical shadows

TL;DR

This paper constructs new non-semisimple quantum invariants for 3-manifolds at roots of unity, enabling the realization of any abelian non-central character as a classical shadow within the skein module framework.

Contribution

It introduces a method to produce maps on the skein module at roots of unity that realize all abelian non-central characters as classical shadows, expanding the scope of quantum invariants.

Findings

Constructed maps on skein modules at roots of unity

Realized all abelian non-central characters as classical shadows

Extended the application of non-semisimple quantum invariants

Abstract

Using the $U_q^Hsl_2$ non-semisimple invariants of 3-manifolds at odd roots of unity, we construct maps on the Kauffman bracket skein module at roots of unity of order twice an odd number, having any possible abelian non central character as classical shadow.