Advanced Variable Topics in Mathematics - Math 191: Knots and Surfaces

MWF 1-2 in MS 7608

Instructor: Olga Radko, MS 5366.  Office hours: TBA


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Course Overview

This╩ course is a is gentle introduction to Surfaces and Knots. There are no formal prerequisites, but some familiarity with linear and abstract algebra, as well as an ability to visualize objects in three dimensions is useful. The course╩ is╩ assessable to advanced undergraduate students. We will start with elementary graph theory and will discuss planar graphs, paths in graphs, spanning trees, and graph colorings (including the six-color theorem). The second central topic is topology of surfaces and classification of surfaces. The main part of the course is learning about knots. One can imagine a knot as a continuous loop (e.g., made of very thin elastic rubber) in the three-dimensional space. Given a knot, one can ask: is it really knotted? I.e., can it be deformed into a ring (the trivial "knot") without making any cuts? More generally, given two knots, one wants to know whether╩ one of them can be deformed into the other. In order to answer such questions, we must introduce and be able to compute (e.g., numerical and polynomial) knot invariants. In this class, we will study many different invariants of knots and will see how they allow to distinguish knots. Knot theory has many relations to topology, physics, and (more recently!) even the study of the structure of DNA. Some of these╩ connections╩ will be explored in the second part of the class.

Textbook:

D. Farmer and T. Stanford "Knots and Surfaces: A Guide to Discovering Mathematics". This textbook is very different from many others. The problems, or "tasks", in the book are an essential part of the text. They provide an insight into how a mathematical discovery can be made. The tasks suggest making mathematical experiments, formulating and testing conjectures, and giving proofs. We will go through Chapter 1 (Networks), Chapter 2 (Surfaces) and Chapter 3 (Knots) of the book. For knots, we will expand the material by learning about many more knot invariants and related topics (including the Jones' polynomial, the Seifert surface, etc.).

Recommended books:

Some of the topics related to knots are not covered in the class' textbook. There are a lot of different  books on Knot Theory. I list below several books which are perhaps the closest to the topics we will study in class and are available at the UCLA library.

  1. C. Adams "The knot book".
  2. L. Kauffmann "Knots and Physics".
  3. D. Rolfsen "Knots and Links".
  4. V.V. Prasolov, A.B. Sossinsky "Knots, Links, Braids and 3-manifolds" .
  5. C. Livingston "Knot theory".


Homework Assignments

Homework is assigned weekly and is due on Fridays in class. Some homework problems are from the textbook (in this case, they are just referred a number), while some of the homework problems are not. You are encouraged to collaborate on solving homework, but each of you needs to write your own solutions. If a homework problem asks to justify your solution, you need to write a mathematically accurate proof in order to receive full credit.
* Homework 1 (due 4/2/10):
Tasks 1.1.1-1.1.3, 1.2.3, 1.3.1, 1.3.2 (the case of seven vertices only), 1.3.3, 1.3.4, 1.4.1.

Midterm

will take place in class on Wednesday, April 21st .

Class project

You are required to write a final paper which either gives a detailed account of some aspect of knot theory (or graph theory, or topology of surfaces) which was not covered in class, or presents

Grading

is based on Homework (30%), Midterm(30%), and Class project (40%). There is no curve in this class, so that your grade is not affected by the performance of other students in the class. To succeed in the course, you need to master the material covered during the class meetings; to be able to solve homework problems and write up your solutions with mathematical accuracy; start thinking about your project topic early and follow the recommended time-line for working on the project.

Handouts