Kuperberg and Turaev-Viro Invariants in Unimodular Categories
Francesco Costantino, Nathan Geer, Bertrand Patureau-Mirand and, Vladimir Turaev
TL;DR
This paper develops a categorical framework extending Penrose graphical calculus to boundary graphs of handlebodies, unifying Kuperberg and Turaev-Viro 3-manifold invariants within a common setting.
Contribution
It introduces a new categorical approach that generalizes existing invariants, connecting Kuperberg and Turaev-Viro invariants through handlebody boundary graphs.
Findings
Unified categorical setting for 3-manifold invariants
Recovery of Kuperberg invariants from involutory Hopf algebras
Recovery of Turaev-Viro invariants from semi-simple spherical categories
Abstract
We give a categorical setting in which Penrose graphical calculus naturally extends to graphs drawn on the boundary of a handlebody. We use it to introduce invariants of 3-manifolds presented by Heegaard splittings. We recover Kuperberg invariants when the category comes from an involutory Hopf algebra and Turaev-Viro invariants when the category is semi-simple and spherical.
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