## Algebraic & Geometric Topology

### Logarithmic Hennings invariants for restricted quantum ${\mathfrak{sl}}(2)$

#### Abstract

We construct a Hennings-type logarithmic invariant for restricted quantum $s l ( 2 )$ at a  root of unity. This quantum group $U$ is not quasitriangular and hence not ribbon, 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 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4329-4358.

Dates
Accepted: 7 June 2018
First available in Project Euclid: 18 December 2018

https://projecteuclid.org/euclid.agt/1545102073

Digital Object Identifier
doi:10.2140/agt.2018.18.4329

Mathematical Reviews number (MathSciNet)
MR3892247

Zentralblatt MATH identifier
07006393

#### Citation

Beliakova, Anna; Blanchet, Christian; Geer, Nathan. Logarithmic Hennings invariants for restricted quantum ${\mathfrak{sl}}(2)$. Algebr. Geom. Topol. 18 (2018), no. 7, 4329--4358. doi:10.2140/agt.2018.18.4329. https://projecteuclid.org/euclid.agt/1545102073

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