David Stanovský
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| ALGEBRAIC INVARIANTS IN KNOT THEORY 2021/22 |
Syllabus:
- Fundamental concepts of knot theory - equivalence of knots, knot notation, Reidemeister moves
- Basic knot invariants, knot coloring
- Seifert surfaces, Seifert matrix, Alexander polynomial
- Braids
- Skein relations and Jones polynomial
- maybe some applications
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.
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.
Preliminary plan:
| topic | Murasugi | other materials | |
| 18.2. | Introduction: a computational view of knot theory. Fundamental concepts: knots, links, knot equivalence, connected sum and prime decomposition of knots. |
1.1-1.5 | intro slides, full intro |
| 25.2. | Diagrams, knot tables, knot graphs. | 2.1-2.3 | |
| 4.3. | Fundamental problems of knot theory. Some naive invariants. | 3.1-3.2, 4.2-4.4 | |
| 11.3. | Reidemester theorem. Linking number. | 4.1, 4.5 | |
| 18.3. | Knot coloring. | 4.6, -- | more on quandle coloring |
| 25.3. | Quandle coloring. Seifert surfaces and their genus. | --, 5.1, 5.2 | |
| 1.4. | Seifert matrices. | 5.2, 5.3 | |
| 8.4. | Equivalence of Seifert matrices. Alexander polynomial. | 5.4, 6.1, 6.2 | |
| 22.4. | Properties of the Alexander polynomial, the signature of knot. Torus knots (construction). | 6.3, 6.4, 7.1 | read 7.2-7.5 |
| 29.4. | Jones polynomial: definition, calculation. | 11.1, 11.2 | |
| 6.5. | Jones polynomial: properties, generalizations. | 11.3, 11.4 | |
| 13.5. or 27.5. | Braid group, constructing knots from braids. | 10.1-10.4 | |
| 20.5. | Applications or Vassiliev invariants. | 12-14 |
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.
