Lie superalgebra invariants and almost classical knots

Abstract

A virtual link is said to be almost classical (AC) if it has a homologically trivial representative in some thickened surface $\Sigma \times [0,1]$, where $\Sigma$ is a closed orientable surface. AC links provide a useful window for observing the geometric topology of virtual knots. Here we take a different approach and look at AC links through the lens of quantum topology. Two adjustments are needed to the existing theory. First, it is necessary to generalize the definition of AC to include virtual tangles and, in particular, virtual braids. Secondly, to distinguish AC and non-AC tangles, the additional structure of quantum supergroups is required. For each Lie superalgebra $\mathfrak{gl}(m|n)$, we define a pair of $U_q(\mathfrak{gl}(m|n))$ Reshetikhin-Turaev functors $Q^{m|n}$, $\widetilde{Q}^{m|n} \circ Zh$ on framed virtual tangles. Here $Zh$ denotes the Bar-Natan $Zh$ construction. These functors unify the Alexander polynomial (AP) of AC links and the generalized Alexander polynomial (GAP) of all virtual links into a single quantum model: $Q^{1|1}$ recovers the AP of an AC link and for any virtual link $K$, $\widetilde{Q}^{1|1}\circ Zh(K)$ is the 2-variable GAP. However, when $(m,n) \ne (1,1)$, these invariants are generally distinct from the AP and GAP. Furthermore, in contrast to the classical case, they are not determined by $m-n$. For example, there are virtual knots with trivial GAP but nontrivial $U_q(\mathfrak{gl}(2|2))$ and $U_q(\mathfrak{gl}(3|3))$ invariants. Silver and Williams proved that the GAP vanishes on all AC links. Our main result is a generalization of this theorem to almost classical tangles and the $U_q(\mathfrak{gl}(m|n))$ Reshetikhin-Turaev functors. We prove that if $T$ is an almost classical tangle, then $\widetilde{Q}^{m|n}\circ Zh(T)$ is conjugate to $Q^{m|n}(T)$, with conjugation determined by an Alexander numbering of $T$.