  The web of dual gauge theories engineered from a class of toric Calabi-Yau
threefolds is explored. In previous work, we have argued for a triality
structure by compiling evidence for the fact that every such manifold $X_{N,M}$
(for given $(N,M)$) engineers three a priori different, weakly coupled quiver
gauge theories in five dimensions. The strong coupling regime of the latter is
in general described by Little String Theories. Furthermore, we also
conjectured that the manifold $X_{N,M}$ is dual to $X_{N',M'}$ if $NM=N'M'$ and
$\text{gcd}(N,M)=\text{gcd}(N',M')$. Combining this result with the triality
structure, we currently argue for a large number of dual quiver gauge theories,
whose instanton partition functions can be computed explicitly as specific
expansions of the topological partition function $\mathcal{Z}_{N,M}$ of
$X_{N,M}$. We illustrate this web of dual theories by studying explicit
examples in detail. We also undertake first steps in further analysing the
extended moduli space of $X_{N,M}$ with the goal of finding other dual gauge
theories.
