A lower bound for the genus of a knot using the Links-Gould invariant

Abstract

The Links-Gould invariant of links $LG^{2,1}$ is a two-variable generalization of the Alexander-Conway polynomial. Using representation theory of $U_{q}\mathfrak{gl}(2 \vert 1)$, we prove that the degree of the Links-Gould polynomial provides a lower bound on the Seifert genus of any knot, therefore improving the bound known as the Seifert inequality in the case of the Alexander polynomial. As an example, unlike some classical tools such as the Alexander polynomial and Levine-Tristram signature, this new genus bound detects the fact that the Kinoshita-Terasaka and Conway knots have genus greater or equal to 2.