A supergroup series for knot complements

We introduce a three variable series invariant $F_K(y,z,q)$ for plumbed knot complements associated with a Lie superalgebra $sl(2\mathrm{\mid }1)$. The invariant is a generalization of the $sl(2\mathrm{\mid }1)$-series invariant $\widehat{Z}(q)$ for closed 3-manifolds introduced by Ferrari and Putrov and an extension of the two variable series invariant defined by Gukov and Manolescu (GM) to the Lie superalgebra. We derive a surgery formula relating $F_K(y,z,q)$ to $\widehat{Z}(q)$ invariant. We find appropriate expansion chambers for certain infinite families of torus knots and compute explicit examples. Furthermore, we provide evidence for a non semisimple $Spi{n}^c$ decorated TQFT from the three variable series. We observe that the super $F_K(y,z,q)$ itself and its results exhibit distinctive features compared to the GM series.