  Jaeger-Kauffman-Saleur (JKS) identified the Alexander polynomial with the
$U_q(\mathfrak{gl}(1|1))$ quantum invariant of classical links and extended
this to a 2-variable invariant of links in thickened surfaces. Here we
generalize this story for every Lie superalgebra of type $\mathfrak{gl}(m|n)$.
Following Reshetikhin and Turaev, we first define a virtual
$U_q(\mathfrak{gl}(m|n))$ functor for virtual tangles. When $m=n=1$, this
recovers the Alexander polynomial of almost classical knots, as defined by
Boden-Gaudreau-Harper-Nicas-White. Next, an extended $U_q(\mathfrak{gl}(m|n))$
functor of virtual tangles is obtained by applying the Bar-Natan
$Zh$-construction. This is equivalent to the 2-variable JKS-invariant when
$m=n=1$, but otherwise our invariants are new whenever $n>0$. In contrast with
the classical case, the virtual and extended $U_q(\mathfrak{gl}(m|n))$ functors
are not entirely determined by the difference $m-n$. For example, the
invariants from $U_q(\mathfrak{gl}(2|0))$ (i.e. the classical Jones polynomial)
and $U_q(\mathfrak{gl}(3|1))$ are distinct, as are the extended invariants from
$U_q(\mathfrak{gl}(1|1))$ and $U_q(\mathfrak{gl}(2|2))$.
  The JKS-invariant was previously shown to be a slice obstruction for virtual
links. We present computational evidence that each extended
$U_q(\mathfrak{gl}(m|m))$ polynomial is also virtual slice obstructions.
Assuming this conjecture holds for just $m=2$, it follows that the virtual
knots 6.31445 and 6.62002 are not slice. Both these knots have trivial
JKS-invariant, trivial graded genus, trivial Rasmussen invariant, and vanishing
extended Milnor invariants up to high order, and hence, no other slice
obstructions have previously been found.
