Logarithmic link invariants of U‾_q ^H(sl₂) and asymptotic dimensions of singlet vertex algebras

We study relationships between the restricted unrolled quantum group U‾_q ^H(sl₂) at q=e^πi/r, and the singlet vertex operator algebra M(r), r≥2. We use deformable families of modules to efficiently compute (1,1)-tangle invariants colored with projective U‾_q ^H(sl₂)-modules. These invariants relate to the colored Alexander tangle invariants studied in [6,40]. It follows that the regularized asymptotic dimensions of characters of M(r), studied previously by the first two authors, coincide with the corresponding modified traces of open Hopf link invariants. We also discuss various categorical properties of M(r)-mod in connection to braided tensor categories.