2008 POSTECH Summer School on Knots and Manifolds

July 22-25, 2008

Course Abstracts

Seifert matrices and knot invariants

Lecturer: Se-Goo Kim

It is known that every knot is the boundary of an orientable surface, called a Seifert surface. By studying the surface it is possible to learn more about the knot. This lecture discusses geometric and algebraic methods on working with Seifert surfaces. The topics are the classification of surfaces, Seifert surfaces and the genus of a knot, homology theory of surfaces, Seifert matrics and the Alexander polynomial, S-equivalence and the signature of a knot, and concordance of knots.

Braids

Lecturer: Sang Jin Lee

The braid theory plays important roles in several areas in mathematics and has applications to outstanding problems in physics, chemistry and biology. This lecture will introduce basic definitions and results from braid theory. First, we will discuss various methods of defining braid groups such as isotopy classes of strings, the fundamental groups of configuration spacess and mapping class groups of punctured disks. Then, we will study some topics related to knot theory such as the theorems of Alexander and Markov, Burau representation and Alexander polynomial and Temperley-Lieb algebra representation and Jones polynomial.

Presentations of knots

Lecturer: Youngsik Huh

In knot theory various methods to express knots had been devised. Usually a knot presentation induces a certain numeric value that is intrinsic to the knot type. In this talk we will introduce several knot presentations, including polygon, diagram, arc presentation, bridge presentation, and their applications. Also algorithmic aspects to determine intrinsic numeric values are also considered.

3-manifolds

Lecturer: Sangyop Lee and Seungsang Oh

I will introduce some of basic notions and fundamental theorems of 3-manifold topology. I am planning to deal with incompressible surfaces, prime decomposition, Heegaard splitting, weakly reducible splitting and strongly irreducible splitting, Dehn surgery, Kirby move.