#### Volume 18, issue 6 (2018)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
This article is available for purchase or by subscription. See below.
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.

## PDF Access Denied

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/agt

We have not been able to recognize your IP address 18.206.177.17 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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