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, section 6.3, section 6.4
29.4.	Torus knots.	7.1-7.5	chapter 7
6.5.	3-manifolds from knots: Dehn surgery, cyclic cover along knot.	8.1-8.3	chapter 8
13.5.	Tangles and 2-bridge knots.	9.1-9.3	sections 9.1-9.2, comment of 9.3-9.4
20.5.	Braids group, constructing knots from braids.	10.1-10.4	chapter 10
27.5.	Jones polynomial: definition, calculation.	11.1-11.2	sections 11.1-11.2
3.6.	Jones polynomial: properties, generalizations.	11.3-11.4	sections 11.3-11.4

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. There are no fixed dates, we can agree on any date and time by email.
I will ask several questions from different chapters of the book. You must know all essential definitions and theorems, be able to calculate concrete examples. Know all fundamental constructions (such as the Seifert surface and the Seifert matrix). You can skip tricky technical proofs (such as the Reidemeister theorem), but you shall be able to make all routine proofs.
Sample questions: Calculate the value of [... some invariant like coloring, Alex. polynomial, ...] for a concrete knot and show that it really is an invariant. Explain the construction of torus knots and formulate the classification theorem. Draw a 2-bridge knot corresponding to the number 3/5. Describe the braid group and its presentation and tell me some interesting properties.