A lower bound for the genus of a knot using the Links-Gould invariant (Ben-Michael Kohli)

22.02.2024 14:15 – 16:00

For a long time, the Alexander polynomial was the main 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.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-15,Séminaire "Topologie et Géométrie"

entrée libre