Title of the report: Modern Knot Theory in the Historical Perspective

Reporter: Professor Jozef H. Przytycki the George Washington University

Report location: Zoom

School contact: Wang Xiao wangxiaotop@jlu.edu.cn

Report topic 1: Tait's graphs and knots. Adjacency matrix of graphs and Goeritz matrix of knots.

Reporting time: Mar. 4, 2021, 9:00-10:00

Report location: Zoom ID: 865 1101 5111; Passcode: 139374

Summary of the report:

In 1867, Lord Kelvin, motivated by Tait's method of producing vortex smoke rings, came up with the vortex atom theory. He hypothesized that atoms were knotted in a substance called ether. At this time creating a table of the elements was of significant importance to the scientific community, and this theory encouraged Tait to work on the knots classification problem.Tait found an ingenious method of connecting plane graphs and knot or link diagrams and used this in his census of knots and links.We will also discuss two other important concepts in graph theory and knot theory: adjacency (and Laplace) matrices for graphs and related Goeritz matrices for links.

Report topic 2: From Seifert matrix and form to Alexander-Conway polynomial and Tristram-Levine signatures.

Reporting time: Mar. 5, 2021, 9:00-10:00

Report location: Zoom ID: 865 1101 5111; Passcode: 139374

Summary of the report:

We define the Seifert surface via Seifert construction and discuss tube equivalence between them.We introduce a Seifert form (and matrix) on the first homology group of a Seifert surface and show that two such matrices (of forms) for a given link are S- equivalent.We discuss several link invariants coming from a matrix of the Seifert form including determinant, signature, Alexander-Conway polynomial, and Tristram-Levine signature.

Report topic 3: From Fox 3-coloring to Yang-Baxter homology.

Reporting time: Mar. 11, 2021, 9:00-10:00

Report location: Zoom ID: 861 3933 6736; Passcode: 975771

Summary of the report:

We start from naive invariants of arc colorings and survey distributive magmas and their homology with relation to knot theory. We outline potential relations to Khovanov homology and categorification, via Yang-Baxter operators. We use here the fact that Yang-Baxter equation can be thought of as a generalization of self-distributivity. We show how to define and visualize Yang-Baxter homology.

Report topic 4: Introduction to Khovanov homology: from enhanced Kauffman states to applications of the long exact sequence of homology.

Reporting time: Mar. 12, 2021, 9:00-10:00

Report location: Zoom ID: 861 3933 6736; Passcode: 975771

Summary of the report:

The theory of invariants of knots and links was revolutionized with the discovery of the Jones polynomial in May of 1984. One of its most spectacular applications is the proof of the first Tait's conjecture, via the Kauffman bracket polynomial. Soon after its announcement, this development motivated several generalizations such as the HOMFLYPT polynomial.

At the end of the XX century, Mikhail Khovanov (PhD 1997) announced a novel construction of a new and very powerful link invariant: a homology theory categorifying the Jones polynomial, containing more information, and having a richer algebraic structure.

Khovanov homology (KH) offers a nontrivial generalization of the Jones polynomial (and the Kauffman bracket polynomial) of links in $R^3$. A more powerful invariant than the Jones polynomial, this special type of categorification has been extensively developed over the last 20 years detects. In particular, KH detects the unknot which at the moment of writing is still unknown for the Jones polynomial. The gist of KH is that it is a bigraded chain complex associated to a link, in such a way that the homology of the complex is a link invariant.

Furthermore, the graded Euler characteristic of the chain complex is the Jones polynomial, which explains the phrase associated with KH that it categorifies the Jones polynomial. With the idea of achieving an elementary exposition of KH, we present a construction after Oleg Viro (it was his lecture in Gdansk in the summer of 2002 which directed my attention to Khovanov homology).

Brief introduction of the speaker:

Jozef H. Przytycki, Professor at George Washington University, Managing Editor of Journal of Knot Theory and Its Ramifications. Mainly engaged in the research of low-dimensional topology and knot theory, published more than 100 papers in international mathematics journals. Since 1995, the academic conference "Knots in Washington" has been held and 49 sessions have been held so far.