We study relationships between the restricted unrolled quantum group U H q (sl 2 ) at 2r-th root of unity q = e πi/r , r ≥ 2, and the singlet vertex operator algebra M(r). We use deformable families of modules to efficiently compute (1, 1)-tangle invariants colored with projective modules of U H q (sl 2 ). These relate to the colored Alexander tangle invariants studied in [ADO,Mu1]. It follows that the regularized asymptotic dimensions of characters of M(r) 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.

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We construct a direct sum completion C ⊕ of a given braided monoidal category C which allows for the rigorous treatment of infinite order simple current extensions of vertex operator algebras as seen in [CKL]. As an example, we construct the vertex operator algebra V L associated to an even lattice L as an infinite order simple current extension of the Heisenberg VOA and recover the structure of its module category through categorical considerations.