Twisted Kuperberg invariants of knots and Reidemeister torsion via twisted Drinfeld doubles

López Neumann, Daniel

doi:10.1090/tran/9099

Twisted Kuperberg invariants of knots and Reidemeister torsion via twisted Drinfeld doubles
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by Daniel López Neumann;

Trans. Amer. Math. Soc. 377 (2024), 5361-5387

DOI: https://doi.org/10.1090/tran/9099

Published electronically: June 11, 2024

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Abstract:

In this paper, we consider the Reshetikhin-Turaev invariants of knots in the three-sphere obtained from a twisted Drinfeld double of a Hopf algebra, or equivalently, the relative Drinfeld center of the crossed product $\text {Rep}(H)\rtimes \text {Aut}(H)$. These are quantum invariants of knots endowed with a homomorphism of the knot group to $\text {Aut}(H)$. We show that, at least for knots in the three-sphere, these invariants provide a non-involutory generalization of the Fox-calculus-twisted Kuperberg invariants of sutured manifolds introduced previously by the author, which are only defined for involutory Hopf algebras. In particular, we describe the $SL(n,\mathbb {C})$-twisted Reidemeister torsion of a knot complement as a Reshetikhin-Turaev invariant.

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