A supergroup series for knot complements

TL;DR

This paper introduces a new three-variable series invariant for knot complements related to Lie superalgebra $sl(2|1)$, extending previous invariants and providing explicit computations and theoretical insights.

Contribution

It presents a novel three-variable invariant for plumbed knot complements associated with $sl(2|1)$, generalizing prior invariants and exploring its properties and applications.

Findings

Derived a surgery formula relating $F_K (y,z,q)$ to $ ext{Zhat}(q)$.

Computed explicit examples for certain torus knots.

Provided evidence for a non semisimple $Spin^c$ decorated TQFT from the series.

Abstract

We introduce a three variable series invariant $F_K (y,z,q)$ for plumbed knot complements associated with a Lie superalgebra $sl(2|1)$. The invariant is a generalization of the $sl(2|1)$-series invariant $\hat{Z}(q)$ 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 $\hat{Z}(q)$ 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 $Spin^c$ 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.