Date: Tuesday Sept 17 2024


Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)

Title: Introduction to Khovanov Homology

Speaker: Ipsa Bezbarua

Abstract: Developed in 2000 by Mikhail Khovanov while he was studying the representation theory of quantum groups, Khovanov homology has revolutionized the study of knot invariants over the past few decades. In this talk, we will start by looking at the idea of knot invariants and why they are such a powerful tool to understand knots and links. We will first look at a classical invariant, the Jones polynomial. We will then learn the basic definitions involved as well as the formulation of Khovanov homology, its relation to the Jones polynomial as well as a brief sketch of why Khovanov homology is a knot invariant. Finally, we will look at some conjectures about patterns observed in Khovanov homology calculations. This talk is based on Dror Bar-Natan's seminal expository paper "On Khovanov's categorification of the Jones polynomial".


Date: Tuesday Oct 1st, 2024


Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)

Title: Cobordism Categories, TQFTs, and the Chain Complex Perspective on Khovanov Homology

Speaker: Evan Scott (CUNY GC)

Abstract: In Bar-Natan's papers in '05 [BN05] and '07 [BN07], he establishes a perspective on Khovanov homology that has gone on to be very influential in the field: the real invariant coming from this theory isn't the Jones Polynomial, or even the full Khovanov Homology, but the Khovanov chain complex itself, up to chain homotopy. Understanding precisely what this means requires us to discuss categories whose morphisms are (formal sums of) cobordisms and functors from such a category to, say, Q-modules (which are called TQFTs). If there's time after rebuilding the theory, we then discuss the computational advantages of this perspective, namely delooping and Gaussian elimination.

References:


Date: Tuesday Oct 8th, 2024


Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)

Title: Why use Dotted Cobordisms and the Bar-Natan Category?

Speaker: Evan Scott (CUNY GC)

Abstract: We continue the discussion from last week, introducing gradings and the quantum degree. With the theory rebuilt, we'll argue why we should use the Bar-Natan category. This category (and its associated approach) has major advantages from a computational perspective, namely delooping and Gaussian elimination. After introducing these tools, we'll compute the Khovanov homology of the Hopf link live on stage.

References:


Date: Tuesday Oct 29 , 2024


Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)

Title: Tangles and Khovanov Homology

Speaker: Susan Rutter

Abstract: Following on from previous talks discussing the Bar-Natan category, we will discuss how tangles are composed in this category. Time permitting, we will also discuss recent work by Kotelskiy, Watson, and Zibrowius which describes the structure of relevant objects and their morphisms in the Bar Natan category for two-tangles.


Date: Tuesday Nov 19 , 2024


Tuesday, 1:30pm - 2:30 pm in Room 4214.03 (Math Thesis Room)

Title: Khovanov homology's distinguished gift to contact geometry

Speaker: Ipsa Bezbarua (CUNY GC)

Abstract: In 2006, Olga Plamenevskaya introduced a new invariant for transverse knots using Khovanov homology. In contrast to what most researchers were doing, she showed that picking a "distinguished" element from one of the Khovanov homology groups also yields a wealth of information about the knot under study. In this talk, we will first look at the general contact structure on R^3, transverse knots and their classical invariants. Then we will define the distinguished element, look at some of its properties and prove its invariance. Time permitting, we will also show how it relates the self-linking number of a transverse knot to the Rasmussen s-invariant.