Invariants for links and 3-manifolds from the modular category with two simple objects

2023/05/01 by Philipp Korablev, Korablev, Philipp
Computer Science · Mathematics · #57M25 #57M27 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2305.00733

openalex publication_date 2023/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe the simplest non-trivial modular category \mathfrakE with two simple objects. Then we extract from this category the invariant for non-oriented links in 3-sphere and two invariants for 3-manifolds: the complex-valued Turaev - Reshetikhin type invariant tr_ε and the real-valued Turaev - Viro type invariant tv_ε. These two invariants for 3-manifolds are related by the equality |tr_ε|²⋅ (ε + 2) = tv_ε, where ε is a root of the equation ε² = ε + 1. Finally, we show that tv_ε coincides with the well-known ε invariant for 3-manifolds.

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