The universal quantum invariant and colored ideal triangulations

Suzuki, Sakie

doi:10.2140/agt.2018.18.3363

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The universal quantum invariant and colored ideal triangulations

Sakie Suzuki

Algebraic & Geometric Topology 18 (2018) 3363–3402

DOI: 10.2140/agt.2018.18.3363

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 $(2, 3)$ 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

References

Publication

Received: 24 August 2017

Revised: 17 April 2018

Accepted: 30 April 2018

Published: 18 October 2018

Authors

Sakie Suzuki
Department of Mathematical and Computing Science Tokyo Institute of Technology Tokyo Japan

Volume 18, issue 6 (2018)