Problem classes Thursday 9.45-11.15 in 7.527

Lectures start on Monday, 8th of April. In the first two weeks, the Thursday slot will be used for lectures.

This is a master course that also can be taken as an advanced bachelor course.

Prerequisites: Linear Algebra I and II, Algebra I, familiarity with and interest in abstract mathematical structures. Parallely attending Algebra II makes sense, and an introductory course on general representation theory (to be offered in the winter semester 2019/20) can be used to get a broader picture of representation theory and as a starting point for further directions.

Monday, April 8: Knots and links, topological and combinatorial equivalences. Knot diagrams, Reidemeister moves.

Thursday, April 11: Decidability, computability, complexity. Invariants.

Friday, April 12: Kauffman bracket, Tait number (writhe) and Jones polynomial. Example: Hopf link.

Monday, April 15: Geometric braids and braid diagrams. Group structure, generators. Free groups.

Thursday, April 18: Construction of free groups. Presentations. Artin braid group, braid relations.

Friday, April 19: Good Friday.

Monday, April 22: Easter Monday.

Thursday, April 25: Problem class.

Friday, April 26: Pure braids.

Monday, April 29: Closing braids. Markov moves, Markov functions. Representations. Tensor products.

Thursday, May 2: Properties of tensor products.

Thursday, May 2: Burau representation and reduced Burau representation. Subrepresentations and quotient representations.

Friday, May 3: From the reduced Burau representation to a Markov function and the Alexander-Conway polynomial.

Friday, May 3: Modules. Algebras.

Monday, May 6: The Iwahori-Hecke algebra. Statement of main properties. Some facts on symmetric groups. Lemmas.

Thursday, May 9: Problem class.

Friday, May 10: More lemmas. Freeness and basis. Exchange condition and consequences.

Monday, May 13: Comparing H

Thursday, May 16: The HOMFLY-PT polynomial.

Thursday, May 16: Motivation. QYBE and braid relations.

Friday, May 17:

Monday, May 20:

Thursday, May 23: Problem class.

Friday, May 24:

Monday, May 27: Problem class.

Friday, May 31: No lecture.

Monday, June 3:

Problem sheet 1

Problem sheet 2

Problem sheet 3

Adams, The knot book

Burde and Zieschang, Knots

Cromwell, Knots and links

Kauffman, Knots and physics

Murasugi, Knot theory and its applications

Sossinski, Knots

Chari and Pressley, A guide to quantum groups

Humphreys, Reflection groups and Coxeter groups

Jantzen, Lectures on quantum groups

Kassel, Quantum groups

Kassel, Rosso and Turaev, Quantum groups and knot invariants

Kassel and Turaev, Braid groups

(more references to be added)

Pictures of knots, history, ... (University of Wales, Bangor).