Vol. 306, No. 2, 2020

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Kuperberg and Turaev–Viro invariants in unimodular categories

Francesco Costantino, Nathan Geer, Bertrand Patureau-Mirand and Vladimir Turaev

Vol. 306 (2020), No. 2, 421–450
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 semisimple and spherical.

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Keywords
quantum invariants, 3-manifold invariants, Heegaard splitting, modified trace, pivotal category, Kuperberg invariants
Mathematical Subject Classification 2010
Primary: 18D10, 57M27
Milestones
Received: 24 July 2019
Accepted: 7 January 2020
Published: 13 July 2020
Authors
Francesco Costantino
Institut de Mathématiques de Toulouse III - Paul Sabotier
Toulouse
France
Nathan Geer
Department of Mathematics and Statistics
Utah State University
Logan, UT
United States
Bertrand Patureau-Mirand
UMR 6205, LMBA
Université de Bretagne-Sud
Campus Tohannic
Vannes
France
Vladimir Turaev
Department of Mathematics
Indiana University
Bloomington, IN
United States