Knot theory 1000-1M25TW
1. Knots, links, isotopies. Dehn lemma, sphere lemma (without proof). Wirtinger presentation.
2. Linking number. Seifert form. Alexander polynomial as
S-equivalence.
3. Link diagrams. Reidemeister theorem. A quick look onto jet spaces.
4. . Proof of Reidemeister theorem. Loops of Reidemeister moves.
5. Branched covers and their homology. Linking form on cyclic covers.
6.. Infinite cyclic covers. Homology with twisted coefficients. Alexander modules. Twisted Alexander modules.
7.. Blanchfield form. Classification of linking forms over
. Signatures.
8.. 4-genus. Slice knots. Topological versus smooth concordance. First obstructions to concordance.
9. Unknotting number. Algebraic unknotting number. Casson-Gordon invariants.
10. Braids. Braid group. Markov and Alexander theorem with Vogel's proof.
Type of course
Mode
Course coordinators
Learning outcomes
Student can compute algebraic invariants of knots and links.
Student can solve simple theoreitical problems on knot theory
Student understands relations between knot invariants and 3-manifolds
Bibliography
1. Rolfsen, "Knot theory"
2. Livingston, "Introduction to knot theory"
3. Burde Zieschang, "Knot theory"
4. Kawauchi "Surveys on knot theory"
5. Hillman "Algebraic invariants of links, v2"
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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