UC Davis

We analyze the two variable series invariant for knot complements originating from a categorification of the SU(2) WRT invariant of closed oriented 3-manifolds. We are especially interested in examining the conjectured hbar-expansion property and the q-holomonic property of the series invariant through an example of a satellite knot, namely, a cabling of the figure eight knot, which has more than twenty crossings. This cable knot result provides nontrivial evidence for the conjectures and demonstrates the robustness of integrality of the quantum invariant under the cabling operation. Furthermore, we investigate the conjectured relation between the series invariant and the ADO invariants at roots of unity. We reinforce the conjecture by presenting explicit formulas and/or an algorithm for particular ADO invariants of a class of torus knots obtained from the series invariant for complement of a knot. Additionally a one parameter deformation of ADO3 invariants of torus knots is provided, which unifies the three original ADO3 formulas into one formula. We also present a one variable series for closed plumbed 3-manifolds associated with a type I Lie superalgebra osp(2j2) that categorifies the CGP invariant.