David Stanovský
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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 2bridge knots, braids
 Skein relations and Jones polynomial
The fundamental material is the book Knot Theory and Its Applications by Kunio Murasugi, chapters 111. After chapter 4, I will make a short intermezzo on knot coloring, which is the topic of my research.
On Thursdays, 910:30 (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 1.11.5  comment 
introduction chapter 1 
full intro

11.3.  Diagrams, knot tables, knot graphs. 
2.12.3 
chapter 2 

18.3.  Fundamental problems. Reidemeister moves. 
3.13.2, 4.1 
chapter 3 
comment on sec. 4.1 
25.3.  Classical knot invariants. 
4.24.5 
sections 4.24.5 
exercise worksheet 
1.4.  Knot coloring. 
4.6 
4.6 and quandle coloring 
more on quandle coloring 
8.4.  Seifert surfaces and matrices. 
5.15.3 
sections 5.15.3 

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. 
6.26.4 
section 6.2 

29.4.  Torus knots 
7.17.5 


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.
