Logarithmic Link Invariants of ${\bar{U}}_q^H({\mathfrak{s}\mathfrak{l}}_2)$ and Asymptotic Dimensions of Singlet Vertex Algebras

We study relationships between the restricted unrolled quantum group ${\bar{U}}_q^H({\mathfrak{s}\mathfrak{l}}_2)$ at $2r$-th root of unity $q=e^{\pi i/r}, r \geq 2$,andthesingletvertexoperatoralgebra$\mathcal M®$. We use deformable families of modules to efficiently compute $(1,1)$-tangle invariants colored with projective modules of ${\bar{U}}_q^H({\mathfrak{s}\mathfrak{l}}_2)$. These relate to the colored Alexander tangle invariants studied in [ADO, M1]. It follows that the regularized asymptotic dimensions of characters of $\mathcal{M}(r)$ coincide with the corresponding modified traces of open Hopf link invariants. We also discuss various categorical properties of $\mathcal{M}(r)$-mod in connection to braided tensor categories.