Description

We will go over background material, then turn to open problems. Here is the plan:

• Week 1: Basic concepts of knot theory, isotopy, concordance. The Reidemeister moves, examples of knots.
• Week 2: First invariants of knots, three-colorability, n-colorability. Methods for distinguishing knots.
• Week 3: Polynomial invariants I: the Alexander polynomial. Knot genus, its relation to the degree of the Alexander polynomial. The genus of alternating knots. Fibered knots.
• Week 4: Polynomial invariants II: the Jones polynomial. Crossing number, its relation to the Jones polynomial. The crossing number of alternating knots.
• Week 5: The concordance group. Infinitely generated Abelian groups. Recollection of some homological algebra.
• Week 6: Knot Floer homology; computations for (1,1)-knots. Grid presentation of knots, grid homology. The relation to knot Floer homology.
• Week 7: Knot invariants and concordance homomorphisms from knot Floer homology. Further topological methods. Open problems for (1,1)-knots.

We will be using handouts and notes. Reference for the first few lectures can be found in http://web.cs.elte.hu/summerschool/2019/egyeb/ssm2019.pdf (pages 62-78).
In addition, some chapters of Grid homology for knots and links (Ozsvath-Stipsicz-Szabo) (AMS Mathematical Surveys and Monographs, Volume 208) will be used.