3-Alterfolds and Quantum Invariants

TL;DR

This paper introduces 3-alterfolds with embedded surfaces and develops quantum invariants that generalize existing 3-manifold invariants, providing new topological tools and insights into algebraic structures like the Drinfeld center.

Contribution

It defines 3-alterfolds with decorated surfaces and constructs quantum invariants, unifying topological moves and algebraic concepts such as the Drinfeld center within a topological framework.

Findings

Quantum invariants of 3-alterfolds generalize 3-manifold invariants.

Unified proof of equality between Reshetikhin-Turaev and Turaev-Viro invariants.

Topological interpretation of algebraic concepts like the Drinfeld center and Frobenius-Schur indicators.

Abstract

In this paper, we introduce the concept of 3-alterfolds with embedded separating surfaces. When the separating surface is decorated by a spherical fusion category, we obtain quantum invariants of 3-alterfold, which is consistent with many topological moves. These moves provide evaluation algorithms for various presentations of 3-alterfold, e.g. Heegaard splittings, triangulations, link surgeries. In particular, we obtain quantum invariants of 3-manifolds containing surfaces, generalizing those of 3-manifolds containing framed links. Moreover, in this framework, we topologize fundamental algebraic concepts. For instance, we implement the Drinfeld center by tube diagrams as a blow up of framed links. The topologized center leads to a quick proof of the equality between the Reshetikhin-Turaev invariants and the Turaev-Viro invariants for spherical fusion categories. In addition, we topologize half-braiding, $S$-matrix and the generalized Frobenius-Schur indicators, etc. In particular, the latter leads to a quick topological proof of the equivariance of the generalized Frobenius-Schur indicators.