The $E_6$ state sum invariant of lens spaces

In this paper, we calculate the values of the $E_6$ state sum invariants for the lens spaces $L(p,q)$. In particular, we show that the values of the invariants are determined by $p\mathrm{m}\mathrm{o}\mathrm{d}12$ and $q\mathrm{m}\mathrm{o}\mathrm{d}(p,12)$. As a corollary, we show that the $E_6$ state sum is a homotopy invariant for the oriented lens spaces.