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

Abstract

The $E_6$ state sum invariant is a topological invariant of closed $3$–manifolds constructed by using the $6j$–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.