David Stanovský
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| ALGEBRAIC INVARIANTS IN KNOT THEORY 2024/25 |
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 principal material is the book Knot Theory and Its Applications by Kunio Murasugi, chapters 1-7, 10-11, 15. The topics will be slightly reordered and a few things will be added from the new booklet Knots, Links, and their Invariants by Alexei Sossinsky.
As a supplementary material, covid 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):
| lecturer | topic | Murasugi | Sossinsky | other materials | exercises | |
| 1.10. | DS | Introduction: a computational view of knot theory. Fundamental concepts: knots and links, equivalence, diagrams . |
--- 1.1-1.4, 2.1 |
--- 1.1, 1.3, 1.4 |
intro slides pěkné motivační video |
Gauss codes [M 2.2] |
| 8.10. | JŠ | Arithmetic of knots: connected sum, inverses, mirror images. Some naive invariants. |
1.5 4.2-4.4 |
3 4.1-4.3 |
Knot decompositions | |
| 15.10. | JŠ | Reidemester theorem. Invariants: linking number, writhe. | 4.1, 4.5 | 1.2, 8.4 | Reidemeister moves and linking number | |
| 22.10. | DS | Coloring invariants. | 4.6 | more on quandle coloring | Knot coloring | |
| 29.10. | JŠ | Conway polynomial (via skein relation). | 6.2 | 2 | proof of existence | Conway poynomial |
| 5.11. | -- | *** děkanský den *** | -- | |||
| 12.11. | JŠ | Seifert surfaces and their genus, Seifert matrices. | 5.1-5.3 | Seifert surfaces | ||
| 19.11. | JŠ* | Equivalence of Seifert matrices. | 5.4, 6.1 | Alexander polynomial | ||
| 26.11. | JŠ* | Properties of the Alexander polynomial, its relation to the Conway polynomial. | 6.1-6.3 / 6.4 (pract.) | More invariants, distinguisging mirror images | ||
| 3.12. | JŠ* | Kauffmann bracket, Jones polynomial: definition, calculation. | 11.4, 11.1 | 5, 6.1 | Jones polynomial | |
| 10.12. | JŠ | Jones polynomial: properties, generalizations. | 11.2, 11.3 | 6 | ||
| 17.12. | DS | Vassiliev invariants. | 15 | 8-11 | Torus knots | |
| 7.1. | DS | Vassiliev invariants. | 15 |
8-11 | Torus knots |
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
