We investigate a (potentially infinite) series of subfactors, called $3^n$ subfactors, including $A_4$, $A_7$, and the Haagerup subfactor as the first three members corresponding to $n=1,2,3$. Generalizing our previous work for odd $n$, we further develop a Cuntz algebra method to construct $3^n$ subfactors and show that the classification of the $3^n$ subfactors and related fusion categories is reduced to explicit polynomial equations under a mild assumption, which automatically holds for odd $n$.In particular, our method with $n=4$ gives a uniform construction of 4 finite depth subfactors, up to dual,without intermediate subfactors of index $3+\sqrt{5}$. It also provides a key step for a new construction of the Asaeda-Haagerup subfactor due to Grossman, Snyder, and the author.

中文翻译：

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$3^n$子因子的分类及相关融合类别

我们研究了一系列（可能是无限的）子因子，称为 $3^n$ 子因子，包括 $A_4$、$A_7$ 和 Haagerup 子因子作为对应于 $n=1,2,3$ 的前三个成员。概括我们之前对奇数 $n$ 的工作，我们进一步开发了一种 Cuntz 代数方法来构造 $3^n$ 子因子，并表明 $3^n$ 子因子的分类和相关融合类别在温和的假设下被简化为显式多项式方程，它自动适用于奇数 $n$。特别是，我们的 $n=4$ 方法给出了 4 个有限深度子因子的统一构造，最多对偶，没有索引 $3+\sqrt{5}$ 的中间子因子。由于 Grossman、Snyder 和作者，它还为新构建 Asaeda-Haagerup 子因子提供了关键步骤。