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Little is known about Vertex Operator Algebras (VOAs) which are neither semi-simple nor rational, and most of the work on such VOAs has been focused around specific examples such as the Singlet VOAW(2, 2r−1). In this thesis, the relationship between subcategories of the module categories of the Singlet VOA and the unrolled restricted quantum group associated to sl(2), U H q (sl(2)) at 2r-th root of unity q = e πi/r with r ≥ 2 is studied. A family of deformable modules X is used to efficiently compute open Hopf Links for U H q (sl(2)) and particular (1, 1)-tangle invariants colored with projective modules of U H q (sl2). These tangle invariants are extensions of the Alexander invariants defined by Murakami. It is also shown that normalized modified traces of open Hopf links for U H q (sl(2))-modules correspond exactly with the asymptotic quantum dimensions for certain W(2, 2r − 1)-modules.

Logarithmic Hopf Link Invariants for the Unrolled Restricted Quantum Group of sl(2)