$K$-stability of $C^{\ast }$-algebras generated by isometries and unitaries with twisted commutation relations

In this article, we prove $K$-stability for a family of $C^{\ast }$-algebras, which are generated by a finite set of unitaries and isometries satisfying twisted commutation relations. This family includes the $C^{\ast }$-algebra of doubly non-commuting isometries and free twist of isometries. Next, we consider the $C^{\ast }$-algebra $A_{\mathcal{V}}$ generated by an $n$-tuple of $\mathcal{U}$-twisted isometries $\mathcal{V}$ with respect to a fixed $\left(\genfrac{}{}{0px}{}{n}{2}\right)$-tuple $\mathcal{U}=\{U_{ij}:1\le i<j\le n\}$ of commuting unitaries (see \cite{NarJaySur-2022aa}). Under the assumption that the spectrum of the commutative $C^{\ast }$-algebra generated by $(\{U_{ij}:1\le i<j\le n\})$ does not contain any element of finite order in the torus group ${\text{bbbt}}^{\left(\genfrac{}{}{0px}{}{n}{2}\right)}$, we show that $A_{\mathcal{V}}$ is $K$-stable. Finally, we prove the same result for the $C^{\ast }$-algebra generated by a tuple of free $\mathcal{U}$-twisted isometries.