Algebraic & Geometric Topology

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

Anna Beliakova, Christian Blanchet, and Nathan Geer

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Abstract

We construct a Hennings-type logarithmic invariant for restricted quantum s l ( 2 ) at a  2 p  th 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 0  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.

Article information

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

Dates
Received: 10 April 2018
Accepted: 7 June 2018
First available in Project Euclid: 18 December 2018

Permanent link to this document
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

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
quantum invariants links Hopf algebras quantum Groups

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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