Lie Superalgebra generalizations of the Jaeger-Kauffman-Saleur Invariant

Abstract: 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.