# Logarithmic Hennings invariants for restricted quantum sl(2)

## TL;DR

This paper introduces a new logarithmic invariant for 3-manifolds with links, based on a restricted quantum group at roots of unity, extending previous invariants with novel algebraic and topological features.

## Contribution

It constructs a logarithmic invariant for restricted quantum sl(2) that is not braided but factorizable, using universal invariants and modified traces, extending Murakami's invariant.

## Key findings

- Invariant applies to pairs of 3-manifolds and colored links.
- Uses non-braided, factorizable quantum group at roots of unity.
- Extends logarithmic invariants with new algebraic structures.

## Abstract

We construct a Hennings type logarithmic invariant for restricted quantum $\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$ is not braided, but factorizable. The invariant is defined for a pair: a 3-manifold $M$ and a colored link $L$ inside $M$. The link $L$ is split into two parts colored by central elements and by trace classes, or elements in the $0^{\text{th}}$ Hochschild homology of $U$, respectively. The two main ingredients of our construction are the universal invariant of a string link with values in tensor powers of $U$, and the modified trace introduced by the third author with his collaborators and computed on tensor powers of the regular representation. Our invariant is a colored extension of the logarithmic invariant constructed by Jun Murakami.