Introduction to knot theory (2015-16) |

#### Course information:

**Here****is a tentative description of the course content**(most probably subject to changes)

- There are many introductory books on knot theory, and the course is mostly based on the following ones:
- The Knot Book, by C. Adams (AMS)
- Knots and Links, by D. Rolfsen (AMS Chelsea)
- Knots, by G. Burde et H. Zieschang (de Gruyter)
- Introduction to knot theory, by R. Lickorish (Springer)
- Vassiliev knot invariants, by S. Chmutov, S. Duzhin et J. Mostovoy (Cambridge)
- Quantum invariants, by T. Ohtsuki (World Scientific)

- Below is a brief recap of the material covered in each course

#### March 15th, (1:30pm-4:30pm)

Finite type invariants (definition, examples)
; chord diagrams, weight systems and the fundamental theorem

#### March 8th, (1:30pm-4:30pm)

Properties of the Jones polynomial
(orientation, miror image, skein relation, connected sum) and Tait
conjecture ; Quantum invariants (Yang-Baxter equations and braid
representations, enhanced Yang-Baxter operators and link invariants)

#### March 1st, (1:30pm-4:30pm)

Milnor invariants (definition, example,
properties) ; Jones polynomial (definition via Kauffman bracket)

#### Feb. 11th, 9am-12am

Alexander polynomial (definition using the
infinite cyclic cover, computation using Seifert matrices, properties,
Alexander-Conway polynomial) ; Milnor invariants (definition)

#### Feb. 2nd, 1:30pm-4:30pm

linking number (classification results) ; the knot
group (def, case of torus knots, wirtinger presentation, Waldhausen
theorem) ; matrix presentation and order of a module

#### Jan. 26th, 1:30pm-4:30pm

braids ; Alexander theorem ; braid index ; Markov theorem ; linking number (equivalent definitions)

#### Jan. 19th, 2pm-5pm

crossing number ; unknotting number ; Seifert surfaces ; knot genus

#### Jan. 12th, 2pm-4pm

proof of the Reidemeister theorem ; the
tricolorability invariant ; connected sum and the knot monoid

#### Jan. 12th, 10am-12am

Definition of knots and knot
equivalence ; polygonal knots and p-equivalence ; knot diagrams ;
Reidemeister theorem

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