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Expanding upon earlier results [arXiv:1702.02861], we present a compendium of Yang-Baxter $\sigma$-models associated with integrable deformations of AdS$_5$. Each example we study from four viewpoints: conformal Drinfeld twists, closed string gravity backgrounds, open string parameters and dual noncommutative (NC) gauge theory. Irrespective of whether the deformed background is a solution to supergravity or generalized supergravity, we show that the open string metric associated with each gravity background is undeformed AdS$_5$ with constant open string coupling and the NC parameter $\Theta$ is directly related to the conformal twist. One novel feature is that $\Theta$ exhibits ``holographic noncommutativity'': while it may exhibit non-trivial dependence on the holographic direction, \textcolor{red}{its value everywhere in the bulk is uniquely determined by its value at the boundary,} thus facilitating introduction of the dual NC gauge theory. We show that the divergence of the NC parameter $\Theta$ is directly related to the unimodularity of the twist. We discuss the implementation of an outer automorphism of AdS as a coordinate transformation in the bulk and discuss its implications for Yang-Baxter $\sigma$-models and self-T-duality based on fermionic T-duality. Finally, we discuss implications of our results for the integrability of associated open strings and planar integrability of dual NC gauge theories.
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