[2209.07378] Quantum invariants of closed-frame $3$ manifolds based on ideal triangulation

[Submitted on 15 Sep 2022]

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Overview: We construct a new type of quantum invariant for closed-frame $3$ manifolds where the first Betty number vanishes. Invariants are defined for arbitrary finite-dimensional Hopp algebras, such as small quantum groups, and are based on ideal triangulations. It uses Heisenberg double-precision regular elements satisfying the pentagon equation and the graphical representation of the $3$ manifold introduced by R. Benedetti and C. Petronio. The structure is simple and intuitive to understand. The pentagon equation reflects the Pachner (2,3) translation of the ideal triangulation, and the Hopf algebra uncomplexity reflects the framing. For the involuntary Hop algebra, the invariants reduce to the invariants of the closed $3$ manifold. For involuntary unimodular co-unimodular Hopf algebras, the invariants reduce to the topological invariants of closed $3$ manifolds. This has been introduced in a previous paper. In this paper, we use his Hopf monoid of the symmetric pivot category more generally to formalize the configuration and use tensor networks for computation.