Volume 18, issue 6 (2018)

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The universal quantum invariant and colored ideal triangulations

Sakie Suzuki

Algebraic & Geometric Topology 18 (2018) 3363–3402
Abstract

The Drinfeld double of a finite-dimensional Hopf algebra is a quasitriangular Hopf algebra with the canonical element as the universal $R$–matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal $R$–matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang–Baxter equation of the universal $R$–matrix. On the other hand, the Heisenberg double of a finite-dimensional Hopf algebra has the canonical element (the $S$–tensor) satisfying the pentagon relation. In this paper we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant of equivalence classes of colored ideal triangulations of $3$–manifolds up to colored moves. In this construction, a copy of the $S$–tensor is attached to each tetrahedron, and invariance under the colored Pachner $\left(2,3\right)$ moves is shown by the pentagon relation of the $S$–tensor.

Keywords
knots and links, 3-manifolds, Heisenberg double, Drinfeld double, universal quantum invariant, colored ideal triangulation
Mathematical Subject Classification 2010
Primary: 16T25, 57M27, 81R50