Graduate Student Seminar- Khovanov homology and the PS braid conjecture

Fri, 16 November, 2018 7:00pm

Title:Khovanov homology and the PS braid conjecture

Speaker: Sujoy Mukherjee - GWU

Date and Time: Friday, November 16, 2:00-3:00pm

Place: Rome 206

Abstract: In 1984, knot theory was revolutionized with the discovery of the Jones polynomial. Fifteen years later, with several questions about it still unanswered, the polynomial was categorified into what is presently known as Khovanov homology (KH). The idea of KH is to associate a bigraded chain complex to a link whose homology is an invariant of the link itself. Additionally, the Euler characteristic of this chain complex, when interpreted appropriately, is the Jones polynomial.

Parts of the PS braid conjecture state that the order of the torsion subgroups in the KH of a closed braid is less or equal to its braid index. In 2017, with the discovery of links with large even torsion subgroups in their KH, this statement was resolved. At the same time, for the case of odd torsion subgroups, the first infinite families of knots and links with odd torsion subgroups up to $\mathbb{Z}_7$ were introduced.

In this talk, after providing a short introduction to KH, we will focus on knots and links with larger odd torsion subgroups than $\mathbb{Z}_7$ , like $\mathbb{Z}_9$ , $\mathbb{Z}_{27}$ , and $\mathbb{Z}_{25}$ . Additionally, we will discuss other recent developments in the study of torsion in KH.