#### Volume 13, issue 6 (2013)

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The state sum invariant of $3$–manifolds constructed from the $E_6$ linear skein

### Kenta Okazaki

Algebraic & Geometric Topology 13 (2013) 3469–3536
##### Abstract

The ${E}_{6}$ state sum invariant is a topological invariant of closed $3$–manifolds constructed by using the $6j\phantom{\rule{0.3em}{0ex}}$–symbols of the ${E}_{6}$ subfactor. In this paper, we introduce the ${E}_{6}$ linear skein as a certain vector space motivated by ${E}_{6}$ subfactor planar algebra, and develop its linear skein theory by showing many relations in it. By using this linear skein, we give an elementary self-contained construction of the ${E}_{6}$ state sum invariant.

##### Keywords
state sum invariant, Turaev–Viro–Ocneanu invariant, $E_6$ subfactor planar algebra, $3$–manifolds, triangulation, linear skein
##### Mathematical Subject Classification 2010
Primary: 57M27, 57M15
Secondary: 46L37
##### Publication
Revised: 4 June 2013
Accepted: 6 June 2013
Published: 10 October 2013
##### Authors
 Kenta Okazaki Research Institute for Mathematical Sciences Kyoto University Kyoto-shi 606-8502 Japan http://www.kurims.kyoto-u.ac.jp/~junes/