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We show how higher-genus su(N) fusion multiplicities may be computed as thediscretized volumes of certain polytopes. The method is illustrated by explicitanalyses of some su(3) and su(4) fusions, but applies to all higher-point andhigher-genus su(N) fusions. It is based on an extension of the realm ofBerenstein-Zelevinsky triangles by including so-called gluing and loop-gluingdiagrams. The identification of the loop-gluing diagrams is our main newresult, since they enable us to characterize higher-genus fusions in terms ofpolytopes. Also, the genus-2 0-point su(3) fusion multiplicity is found to be asimple binomial coefficient in the affine level.Comment: 21 pages, LaTe

Higher-genussu(N) fusion multiplicities as polytope volumes