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

Abstract

For a long time, the Alexander polynomial was the only easily computable link invariant to be known. But in 1984, Jones discovered his well known polynomial link invariant, and that gave birth to the vast theory of quantum link invariants. However, unlike for the Alexander invariant, it is in general hard to deduce precise topological properties on a knot or link from the value quantum invariants take on that link. For instance, no genus bound is known for the Jones polynomial. The Links-Gould invariants of oriented links $LG^{m,n}(L,t_{0},t_{1})$ are two variable quantum invariants obtained by the Reshetikhin-Turaev construction applied to Hopf superalgebras $U_{q}\mathfrak{gl}(m \vert n)$. These invariants are known to be generalizations of the Alexander invariant. Using representation theory of $U_{q}\mathfrak{gl}(2 \vert 1)$, we proved in recent work with Guillaume Tahar that the degree of the Links-Gould polynomial $LG^{2,1}$ 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 invariant. Reference: arXiv:2310.15617

Speaker Intro

I was born in France and grew up between France and the United States. I studied math at ENS de Cachan (Paris), Université Paris 7 and Université de Bourgogne, where I obtained my PhD (directed by P. Schauenburg and E. Wagner). My mathematical interests are related to low dimensional topology. I have been studying knot and link theory, and more precisely connections that exist between classical and quantum link invariants. On a more personal level, I enjoy spending time with my three children Côme, Aliocha and Madeleine.