Planar algebras and Kuperberg's 3-manifold invariant

2005/09/14 by Vijay Kodiyalam, Kodiyalam, Vijay, V. S. Sunder +1
Mathematics · #16W30 #57M27 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #math.OA #math.QA #msc:16W30 #msc:57M27

paper · pdf · doi:10.48550/arxiv.math/0509302

19 pages, 9 figures

arxiv created 2005/09/14 · openalex publication_date 2005/09/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We recapture Kuperberg's numerical invariant of 3-manifolds associated to a semisimple and cosemisimple Hopf algebra through a `planar algebra construction'. A result of possibly independent interest, used during the proof, which relates duality in planar graphs and Hopf algebras, is the subject of a final section.

Related

• Invariants of 3-Manifolds Derived From Finite Dimensional Hopf Algebras
• A New Proof of \bfZ_Kup=|Z_Henn|² for Semisimple Hopf Algebras
• On a Universal Invariant of 3-Manifolds
• On Two Proofs for the Existence and Uniqueness of Integrals for Finite-Dimensional Hopf Algebras
• Semisimple Hopf Algebras via Geometric Invariant Theory