Using the methods developed in a previous paper, we generalize the calculation of the dual amplitude for a nonplanar diagram with a single closed loop to the case with an arbitrary number of "twisted vertices." Just as with the four-point function discussed in the previous paper, we find that, if we write the amplitude $M=\int {}_{\Sigma }{}^{}{}_{}{}^{}d_{}^4{}_{}{}^{}\Pi \mathfrak{M}\left(\Pi \right)$, $\mathfrak{M}\left(\Pi \right)$ is periodic in $\Pi$, the period being given by the sum of the four-momenta of the "twisted vertices." We then show in the general case that the prescription of choosing $\Sigma$ to range over just one period yields the imaginary part required by perturbative unitarity. We verify that $M$ so defined is dual in three different ways. We show explicitly that our result is equivalent to the Kikkawa-Klein-Sakita-Virasoro (KKSV) prescription; we also prove duality directly from the Bardakci-Ruegg-like form without reference to the KKSV structure; and finally we show that duality is manifest within the operator formalism before the trace is performed.

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