David Stanovský    //   

ALGEBRAIC INVARIANTS IN KNOT THEORY 2026/27

Here is the old version of the course. This year, we plan to focus much more on braid groups.

Syllabus:

  • Fundamental concepts of knot theory: equivalence, Reidemeister moves, basic invariants
  • Coloring invariants
  • Seifert surfaces, Alexander polynomial
  • Skein relations, Conway and Jones polynomial
  • Braid groups

The principal reading material is the book Knot Theory and Its Applications by Kunio Murasugi. The knot theoretic part of the lecture mostly follows the book, up to a couple of reorderings and proof corrections :-) Other useful literature:

  • mathematically heavier approach: W. B. R. Lickorish: An intro to Knot Theory
  • many good semi-popular books on knots, for instance, P. Cromwell: Knots and Links, or C. Adams: The Knot Book
  • knot coloring: Nelson, Elhamdadi: Quandles: An Introduction to the Algebra of Knots
For braid groups, the principal material is the book Lecture Notes on Knot Invariants by Weiping Li. Another good reading is the bachelor thesis of Jan Frnka.

Instructors: Radek Olšák, David Stanovský

Program: (actual for past lectures, tentative for future lectures):

lecturer topic Murasugi other materials exercises
1.10.DS+RO Introduction: a computational view of knot theory.
Fundamental concepts: knots and links, equivalence, diagrams .		 ---
1.1-1.4, 2.1		 intro slides
pěkné motivační video 		Gauss codes [M 2.2]
8.10. RO Arithmetic of knots: connected sum, inverses, mirror images.
Some naive invariants.
 1.5
4.2-4.4 		Knot decompositions
15.10. DS Reidemester theorem. Invariants: linking number, writhe. 4.1, 4.5 Reidemeister moves and linking number
22.10.DS Coloring invariants. 4.6 more on quandle coloring Knot coloring
29.10. DS Knots vs. braids, the braid group. M 10.1-10.4, Li 2.1-2.2
5.11. DS Artin's theorem. Li 2.3
12.11. DS Normal form of braid words. Li 2.4
19.11. DS ** something on braid groups **
26.11. RO Conway polynomial (via skein relation). 6.2 Lickorish, chap. 15 Conway polynomial
3.12. RO Seifert surfaces and their genus, Seifert matrices. 5.1-5.3 Seifert surfaces
10.12. RO Alexander polynomial, relation to Conway. 6.1-6.3 / 6.4 (pract.) Alexander polynomial
(Extra material: More invariants from Seifert, distinguishing mirrors)
17.12. RO Kauffmann bracket, Jones polynomial: definition, calculation. 11.4, 11.1 Jones polynomial
7.1. RO Jones polynomial: properties, generalizations. 11.2, 11.3, 11.4 existence proof: Lickorish, chap. 15 Torus knots and their Jones polynomial

As a supplementary material, covid videos from 2021 are available at the older website of the course. This year, we will cover similar, but not exactly the same topics, and the exposition will be much better.

Exam: The exam is oral. You wil be given two questions:
1. Theory: explain certain topic. Example: "Define the Alexander polynomial using Seifert matrices and show that it is symmetric for knots." Example: "Explain the Reidemester theorem, prove it, and on a concrete example explain how to use it to prove invariance."
2. Calculation: calculate certain invariant of a concrete knot. Example: "Find all 5-colorings of the 9_7 knot (picture will be given)."
There are no fixed dates for the exam. Suggest a date, we will tell you if we are available - at least a couple days ahead of your planned exam. We are in Prague most of the time, but not always free.