David Stanovský
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| ALGEBRAIC INVARIANTS IN KNOT THEORY 2023/24 |
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
- Vassiliev invariants
The fundamental material is the book Knot Theory and Its Applications by Kunio Murasugi, chapters 1-7 and 10-11, possibly a little bit from 12-14. After chapter 4, I will make a short intermezzo on knot coloring, which is the topic of my research. I will also slightly reorder the material and add a few things from the new booklet Knots, Links, and their Invariants by Alexei Sossinsky.
As a supplementary material, videos from 2021 are available at the older website of the course. This year, we will cover similar, but not exactly the same topics.
Program: (actual for past lectures, tentative for future lectures):
| topic | Murasugi | Sossinsky | other materials | exercises | |
| 2.10. | Introduction: a computational view of knot theory. Fundamental concepts: knots and links, equivalence, diagrams, knot tables. |
--- 1.1-1.4, 2.1 |
--- 1.1, 1.3, 1.4 |
intro slides pěkné motivační video |
Gauss codes [M 2.2] |
| 9.10. | Arithmetic of knots: connected sum, inverses, mirror images. Some naive invariants. |
1.5 4.2-4.4 |
3 4.1-4.3 |
Prime decomposition and naive invariants | |
| 16.10. | Reidemester theorem. Invariants: linking number, writhe. | 4.1, 4.5 | 1.2, 8.4 | Reidemeister moves and linking number. | |
| 23.10. | Coloring invariants. | 4.6 | more on quandle coloring | Calculating knot coloring. | |
| 30.10. | Conway polynomial (via skein relation). | 6.2 | 2 | Calculating the Conway polynomial. | |
| 6.11. | Seifert surfaces and their genus, Seifert matrices. | 5.1-5.3 | Calculating Seifert surfaces. | ||
| 13.11. | Equivalence of Seifert matrices. | 5.4, 6.1 | Seifert surfaces and genus. | ||
| 20.11. | Properties of the Alexander polynomial, its relation to the Conway polynomial. | 6.1-6.3 / 6.4 (pract.) | Signature of a knot, and distinguishing mirror images. | ||
| 27.11. | Kauffmann bracket, Jones polynomial: definition, calculation. | 11.5, 11.1, 11.2 | 5, 6.1 | Calculating the Jones polynomial. | |
| 4.12. | Jones polynomial: properties, generalizations. | 11.3, 11.4 | 6 | Calculating the Jones and other polynomials. | |
| 11.12. | Braid group, constructing knots from braids. | 10.1-10.4 | 7 | Braid groups. | |
| 18.12. | Vassiliev invariants. | 14 | 8-11 | Torus knots: definition. | |
| 1.1. | --- | Torus knots: equivalence. | |||
| 8.1. | Vassiliev invariants. | 14 | 8-11 | Torus knots: the Jones polynomial. |
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: The exam is oral. You wil be given two questions:
1) A long one, asking to explain certain topic. Example: "Define the Alexander polynomial using Seifert matrices, explain why it is an invariant and show that it is symmetric for knots." Example: "Explain the Reidemester theorem and prove it."
2) A short one, asking to calculate certain invariant. Example: "Find 5-coloring of a given knot (concrete picture will be given)."
I expect you to know all proofs and methods as explained at the lecture. In particular, if the proof was just a sketch, you are expected to understand the idea, but I will not ask more details than presented at the lecture.
There are no fixed dates for the exam. Suggest a date, I will tell you if I am available - at least a couple days ahead of your planned exam. I am in Prague most of the time, but I am not always free - I have large scale exams on general algebra, occassional meetings, etc.
Following a tradition, there is one exam date in SIS, and it is special: exam in nature (zkouška v přírodě). See SIS for a description and feel free to ask for more info.
