David Stanovský    //   

ALGEBRAIC INVARIANTS IN KNOT THEORY 2020/21

Syllabus:

  • Fundamental concepts of knot theory - equivalence of knots, knot notation, Reidemeister moves
  • Basic knot invariants, knot coloring
  • Seifert surfaces, Seifert matrix, Alexander polynomial
  • Dehn surgery and covering spaces, tangles and 2-bridge knots, braids
  • Skein relations and Jones polynomial

The fundamental material is the book Knot Theory and Its Applications by Kunio Murasugi, chapters 1-11. After chapter 4, I will make a short intermezzo on knot coloring, which is the topic of my research.

On Thursdays, 9-10:30 (or rather a small fraction of it), we will meet on zoom to discuss the book. I will present a brief overview of the next topic to read, and we can discuss unclear parts of the previous topic. We can use the Friday slot (or any other time) for further discussions, but most of the time devoted to the course shall be spent by reading the book.

Reading plan:

topic Murasugi lecture video other materials
4.3.Introduction: a computational view of knot theory.
Fundamental concepts: knots, links, knot equivalence, connected sum and prime decomposition of knots.
intro slides
1.1-1.5 - comment
introduction
chapter 1
full intro
11.3.Diagrams, knot tables, knot graphs. 2.1-2.3 chapter 2
18.3.Fundamental problems. Reidemeister moves. 3.1-3.2, 4.1 chapter 3 comment on sec. 4.1
25.3.Classical knot invariants. 4.2-4.5 sections 4.2-4.5 exercise worksheet
1.4.Knot coloring. 4.6 4.6 and quandle coloring more on quandle coloring
8.4.Seifert surfaces and matrices. 5.1-5.3 sections 5.1-5.3
15.4.Equivalence of Seifert matrices. Alexander polynomial. 5.4, 6.1, 6.2 section 5.4, section 6.1, section 6.2
22.4.Properties of the Alexander polynomial, the signature of knot. 6.2-6.4 section 6.2
29.4.Torus knots 7.1-7.5

Other literature:

  • any book on knot theory, a particularly good one is, 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

Exam: oral exam, topics covered by the lecture. Details will be announced later.