MATH2141 and MATH2151
Lecturer : Dirk Schuetz
Term : Epiphany 2011
Lectures :Problem Classes : Friday 10:00 in CG 85 (odd weeks only)
Literature
Here is a set of notes.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 | |
Problem Set 2 pdf | 2,4 | - | 07.02.11 | |
Problem Set 3 pdf | 6,8b,10 | 1,2,3,4,7a,7c,8a | 14.02.11 | |
Problem Set 4 pdf | 1a,2 | - | 21.02.11 | |
Problem Set 5 pdf | 2,5,6 | 1,3 | 28.02.11 | |
Problem Set 6 pdf | 2,3 | - | 07.03.11 | |
Problem Set 7 pdf | 2,5,6b,6d | 1,4,6a,6c,6e | 14.03.11 | |
Problem Set 8 pdf | - | - | - | |
Problem Set 9 pdf | - | - | - |
Links
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. |