# 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.