Let \(m, n \in \mathbb {N}_0\) , and let X be a closed subset of \(\mathbb {T}^{\left( {\begin{array}{c}m+n\\ 2\end{array}}\right) }\) . We define \(C^{m,n}_X\) to be the universal \(C^*\) -algebra among those generated by m unitaries and n isometries satisfying doubly twisted commutation relations with respect to a twist \(\mathcal {U} = \left\{ U_{ij}\right\} _{1 \le i < j \le m+n}\) of commuting unitaries having joint spectrum X. We provide a complete list of the irreducible representations of \(C^{m,n}_X\) up to unitary equivalence and, under a denseness assumption on X, explicitly construct a faithful representation of \(C^{m,n}_X\) . Under the same assumption, we also give a necessary and sufficient condition on a fixed tuple \(\mathcal {U}\) of commuting unitaries with joint spectrum X for the existence of a universal tuple of \(\mathcal {U}\) -doubly twisted isometries. For \(X = \mathbb {T}^{\left( {\begin{array}{c}m+n\\ 2\end{array}}\right) }\) , we compute the K-groups of \(C^{m,n}_X\) . We further classify the \(C^*\) -algebras generated by a pair of doubly twisted isometries with a fixed parameter \(\theta \in \mathbb {R} {\setminus } \mathbb {Q}\) , whose wandering spaces are finite-dimensional. Finally, for a fixed unitary U, we classify all the \(C^*\) -algebras generated by the unitary U and a pair of U-doubly twisted isometries with finite-dimensional wandering spaces.
