Lie Superalgebra generalizations of the Jaeger-Kauffman-Saleur Invariant

Jaeger-Kauffman-Saleur (JKS) identified the Alexander polynomial with the Uq (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 gl (m|n). Following Reshetikhin and Turaev, we first define a virtual Uq (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 Uq (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 Uq (gl (m|n) ) functors are not entirely determined by the difference m-n. For example, the invariants from Uq (gl (2|0) ) (i. e. the classical Jones polynomial) and Uq (gl (3|1) ) are distinct, as are the extended invariants from Uq (gl (1|1) ) and Uq (gl (2|2) ). The JKS-invariant was previously shown to be a slice obstruction for virtual links. We present computational evidence that each extended Uq (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.