Volume 18, issue 7 (2018)

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

Anna Beliakova, Christian Blanchet and Nathan Geer

Algebraic & Geometric Topology 18 (2018) 4329–4358

We construct a Hennings-type logarithmic invariant for restricted quantum sl(2) at a 2p 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.

quantum invariants, links, Hopf algebras, quantum Groups
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 17B37, 57M25
Received: 10 April 2018
Accepted: 7 June 2018
Published: 11 December 2018
Anna Beliakova
Institut für Mathematik
Universität Zürich
Christian Blanchet
Institut de Mathématiques de Jussieu - Paris Rive Gauche
Université Paris 7 (Denis Diderot)
Nathan Geer
Department of Mathematics and Statistics
Utah State University
Logan, UT
United States