$U_q(\mathfrak{gl}(m|n))$ bounds on the minimal genus of virtual links

Abstract

For links $L \subset \Sigma \times [0,1]$, where $\Sigma$ is a closed orientable surface, we define a $U_q(\mathfrak{gl}(1|1))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(\Sigma)]$. This invariant turns out to be equivalent to an infinite cyclic version of the Carter-Silver-Williams (CSW) polynomial. The importance of the CSW polynomial is that half its symplectic rank gives strong lower bounds on the virtual genus. Recall that the virtual genus of a virtual link $J$ is the smallest genus of all closed orientable surfaces $\Sigma$ on which $J$ can be represented by a link diagram on $\Sigma$. Here we generalize the CSW lower bound to all quantum supergroups $U_q(\mathfrak{gl}(m|n))$ with $m,n>0$. For $(m,n)=(1,1)$, the $U_q(\mathfrak{gl}(m|n))$ bound is the same as the CSW bound. However, changing the value of the pair $(m,n)$ can give lower bounds better than those available from other known methods. We compare the $U_q(\mathfrak{gl}(m|n))$ lower bounds to those coming from the CSW polynomial, the surface bracket, the arrow polynomial, hyperbolicity, and the Gordon-Litherland determinant test. As a first application, we show that the Seifert genus of homologically trivial knots in thickened surfaces is not additive under the connected sum operation of virtual knots. As a second application, we prove that the Jaeger-Kauffman-Saleur invariant of a virtual knot is always realizable as the Alexander polynomial of an infinite cyclic cover of a knot complement in some $\Sigma \times [0,1]$, but is not always so on a surface of minimal genus. This is accomplished with a generalization of the $Zh$-construction, called the homotopy $Zh$-construction.