Geometric Topology

MATH2141 and MATH2151

Lecturer : Dirk Schuetz

Term : Epiphany 2011

Lectures :
  • Thursday 15:15 in CG 85
  • Friday 15:15 in E 005

Problem Classes : Friday 10:00 in CG 85 (odd weeks only)

Literature

Here is a set of notes.

The following is a list of books on which the lecture is based. Although we will not follow a book strictly, the material can be found in them and they may sometimes offer a different approach.
  • C.C. Adams, The Knot Book. American Mathematical Society, Providence, RI, 2004.
  • N.D. Gilbert and T. Porter, Knots and Surfaces. Oxford Science Publications, Oxford University Press, 1994.
  • V.G. Boltyanskii and V.A. Efremovich, Intuitive Combinatorial Topology. Springer Verlag, New York, 2001.
  • P. Cromwell, Knots and Links. Cambridge University Press, Cambridge, 2004.

Assignments

Problems Homework questions Tutorial questions Hand in on Solutions
Problem Set 1 pdf 2,8,9 5,6,7,3,4 31.01.11 pdf
Problem Set 2 pdf 2,4 - 07.02.11 pdf
Problem Set 3 pdf 6,8b,10 1,2,3,4,7a,7c,8a 14.02.11 pdf
Problem Set 4 pdf 1a,2 - 21.02.11 pdf
Problem Set 5 pdf 2,5,6 1,3 28.02.11 pdf
Problem Set 6 pdf 2,3 - 07.03.11 pdf
Problem Set 7 pdf 2,5,6b,6d 1,4,6a,6c,6e 14.03.11 pdf
Problem Set 8 pdf - - - pdf
Problem Set 9 pdf - - - pdf


Links

  • Knot Atlas
  • Seifert View

Lecture Outline

Date Outline
20.01. In this lecture we will introduce knots and links and discuss isotopies of knots and links, in particular Reidemeister moves.
21.01. In this lecture we will see how Reidemeister moves can be used to pass from one diagram to another, and we will introduce tricolourability in order to distinguish certain knots.
27.01. In this lecture we will see that the figure 8-knot, the Hopf link and the Borromean rings are not tricolourable. We will also see composition of knots.
28.01. In this lecture we will define alternating knots and the crossing number. We will also see some basic properties of the crossing number.
03.02. In this lecture we will introduce the writhe of a link diagram and the linking number of an oriented link. We will use this to distinguish certain links.
04.02. In this lecture we will introduce the bracket polynomial of a link diagram. We will explore its behaviour under Reidemeister moves and obtain a powerful link invariant.
10.02. In this lecture we will calculate the X-polynomial of the trefoil, and distinguish the left- from the right-hand trefoil. We will also define the Jones polynomial.
11.02. In this lecture we will introduce the Alexander-Conway polynomial and the absolute polynomial. We will also see some basic calculations.
17.02. In this lecture we will define surfaces, and give plenty of examples of surfaces.
18.02. In this lecture we will define simplicial complexes, which will give us a combinatorial way to deal with surfaces.
24.02. In this lecture we will define the Euler characteristic and give examples on how to calculate it. We will also define surfaces with boundary and see some examples.
25.02. In this lecture we will discuss orientable surfaces, and state the Classification Theorem of surfaces. We will then show how to identify surfaces and begin with Seifert surfaces.
03.03. In this lecture we will study the Seifert algorithm, calculate examples and show how to identify the surfaces.
04.03. In this lecture we will look at the genus of a composition of knots. We will also define the winding number of a map from the circle to the circle.
10.03. In this lecture we will show how to calculate winding numbers of polynomials, define vector fields, singularities of vector fields and their index.
11.03. In this lecture we will see more examples of indices of singularities, and see how winding numbers behave if circles contain more than one singularity.
17.03. In this lecture we will consider vector fields on spheres, and in particular prove the Hairy-Ball-Theorem.
18.03. In this lecture we will state and prove the Poincaré-Hopf Theorem.

Last modified: 06.05.2011.