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

Understand

We study relationships between the restricted unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{\pi i/r}, r \geq 2$, and the singlet vertex operator algebra $\mathcal M(r)$.

• We use deformable families of modules to efficiently compute $(1, 1)$-tangle invariants colored with projective modules of $\overline{U}_q^H(\mathfrak{sl}_2)$.
• 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.