Monday, 24 August 2020

Andrei Vesnin
Title: --
Abstract:

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Mohamed Elhamdadi
Title: A survey of quandle theory.
Abstract:

Quandles are generally non-associative algebraic structures whose axioms are motivated by Reidemeister moves on knot diagrams.  We start by reviewing quandles and then explain their cohomology theories and some knot theory applications.  Precisely, we will show how 2-cocycles and 3-cocycles can be used to give quandle cocycle invariants of knots in 3-space and knotted surfaces in 4-space respectively. Quandles rings have been investigated recently in parallel to the classical theory of group rings by Bardakov-Passi-Singh.  We will discuss some of their works on this subject.  The talks will be accessible to students.

Louis Kauffman
Title: Introduction to Virtual Knot Theory
Abstract:

These talks will introduce virtual knot theory, its relationships with classical knot theory, with knotoids and with topological graph theory. We will discuss invariants of virtual knots including the bracket polynomial, the arrow bracket polynomial, the Manturov parity bracket, the affine index polynomial and Khovanov homology for virtual knots and links. We will also discuss virtual link cobordism and its relationships with these invariants.

Tuesday, 25 August 2020

Andrei Vesnin
Title: --
Abstract:

--

Mohamed Elhamdadi
Title: A survey of quandle theory
Abstract:

Quandles are generally non-associative algebraic structures whose axioms are motivated by Reidemeister moves on knot diagrams.  We start by reviewing quandles and then explain their cohomology theories and some knot theory applications.  Precisely, we will show how 2-cocycles and 3-cocycles can be used to give quandle cocycle invariants of knots in 3-space and knotted surfaces in 4-space respectively. Quandles rings have been investigated recently in parallel to the classical theory of group rings by Bardakov-Passi-Singh.  We will discuss some of their works on this subject.  The talks will be accessible to students.

Louis Kauffman
Title: Introduction to Virtual Knot Theory
Abstract:

These talks will introduce virtual knot theory, its relationships with classical knot theory, with knotoids and with topological graph theory. We will discuss invariants of virtual knots including the bracket polynomial, the arrow bracket polynomial, the Manturov parity bracket, the affine index polynomial and Khovanov homology for virtual knots and links. We will also discuss virtual link cobordism and its relationships with these invariants.

Wednesday, 26 August 2020

Valeriy Bardakov
Title: --
Abstract:

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Jozef H. Przytycki
Title: --
Abstract:

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Thursday, 27 August 2020

Valeriy Bardakov
Title: --
Abstract:

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Seiichi Kamada
Title: Braids, I
Abstract:

This is an introduction to braids in knot theory. It includes, geometric and algebraic definitions of braids, the pure braid group and representations of the braid group to the permutation group, a graphical method  of computing braid words, Artin automorphisms on the free group, racks and quandles and their automorphisms induced by braids. 

Abhijit Champanerkar
Title: Hyperbolic knot theory
Abstract:

In this series of two talks we will introduce ideas, tools and examples in hyperbolic knot theory. In the first talk we will review basic hyperbolic geometry in dimensions two and three, important structure theorems for hyperbolic 3-manifolds and ideal tetrahedra which are building blocks for hyperbolic 3-manifolds. In the second talk we will discuss ideal triangulations, gluing equations, Thurston's Dehn surgery Theorem and give explicit examples of hyperbolic knots and links. We will also see computational tools like SnapPy to study and compute geometric invariants of hyperbolic knots and 3-manifolds.

Jozef H. Przytycki
Title: --
Abstract:

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Friday, 28 August 2020

Seiichi Kamada
Title: Braids, II
Abstract:

This is an introduction to braids in knot theory. It includes, Alexander and Markov theorems: how knots can be presented by braids, computation of knot groups from braids, computation of knot quandles from braids, quandle cocycle invariants, 2-dimensional braids.

Abhijit Champanerkar
Title: Hyperbolic knot theory
Abstract:

 In this series of two talks we will introduce ideas, tools and examples in hyperbolic knot theory. In the first talk we will review basic hyperbolic geometry in dimensions two and three, important structure theorems for hyperbolic 3-manifolds and ideal tetrahedra which are building blocks for hyperbolic 3-manifolds. In the second talk we will discuss ideal triangulations, gluing equations, Thurston's Dehn surgery Theorem and give explicit examples of hyperbolic knots and links. We will also see computational tools like SnapPy to study and compute geometric invariants of hyperbolic knots and 3-manifolds.