MATH 139
TOPICS IN KNOT THEORY
- Class time: Mondays and Wednesdays 4-5.30pm.
- Location: 110 Science Center
- Office Hours: Monday 2-3pm, Wednesday 2-4pm also by
appointment.
- Email: denne@math.harvard.edu
- Course Webpage: index.html
- Class Assistant: Gerardo Con Diaz
- email: condiaz@fas.harvard.edu
- Section time and location: Monday 9-10pm in SC101B
- Gerardo's Office Hour: TBA
- Announcements:
- Homework 4 is now posted and is due Thursday March 8th.
- Class will be held on Friday 2nd March from 2-3pm in SC 221.
- Everyone should try to meet with me either this week or next week (by March 9) to decide on a reading project.
- Summary: An introduction and overview of knot theory. Each
student will also read about a part of knot theory in depth, give a talk and
write an essay on it.
- Prerequisite: Familiarity with groups (as with Math 101
or Math 122), basic topology (as in Math 131), or consent of
instructor. In particular, you must meet with me if you have
not taken Math 131.
- Texts: Knots and Links by Peter Cromwell (Cambridge). Available
in the Coop or online.
Other excellent sources include:
- Knot Theory by Charles Livingston (MAA);
- Knots and Surfaces by Gilbert and Porter (Oxford);
- Reference: Knots by G. Burde and H. Zieschang (Walter de Gruyter);
- Reference: Knots and Links by Dale Rolfsen (AMS);
- Reference: A survey of Knot Theory by Akio Kawauchi.
The required text and the first two sources can be found on reserve in the
Cabot library. Some of these books are also in the Birkoff library on the
third floor of the Mathematics Department. At some point you'll want to
take a look at the knot theory section in Cabot. (I also own many books on
knot theory.)
- Examination policy: no exams.
- Grading policy: weekly homework assignments: 50%, final
project: 50%. The final project consists of preparation: 10%, talk: 15%,
essay: 25%.
- Final project: A chance to read in depth about a part of knot
theory that interests you. You will write an
essay about your project. You will also give a 20 minute talk to the class
about your topic. This should be at the level of the math table talks. The
preparation grade will come from your meeting with me throughout the
semester as you work on your project. Complete details about the final
project can be found
HERE.
- Homework policy: weekly assignments.
- The assignments and the
dates they are due will be posted on the course website.
- Homework should be turned in the course mailbox (outside 325 Science
Center) by noon on the day that the assignment is due.
- Late homework will be accepted only in special circumstances and
only with prior approval.
- It is OK to discuss the problems amongst yourselves. However each
student must hand in their own solutions that they have written
themselves. (Copying someone else's homework is unacceptable.)
To make the job of grading easier, could
you please follow the following guidelines for homework:
- Write your name on your HW.
- Neat, legible handwriting. We will not grade anything we cannot read!!!
- Write on one side of the paper only.
- The problems should be in the order assigned.
- Staple (or paper-clip) all pages together.
- Attendance: Attendance will not be taken at each
class. However, there is no much point in signing up for this course unless you plan to come to each class and participate in it!
- Drop Date: 26th February, 2007.
- Class handouts:
- Helpful web pages and references:
- Visualising Seifert Surfaces (thanks to Andrew Laitman): http://www.win.tue.nl/~vanwijk/seifertview/
- Table of Knot invariants: http://www.indiana.edu/~knotinfo/
- The knot atlas:
http://katlas.math.toronto.edu/wiki/Main_Page
- Knot plot: http://knotplot.com/
- Joan Birman's suggested reading list: Click here for
.pdf file.
- Survey article: R.H. Fox "A quick trip though knot theory",
Topology of 3-manifolds (M.K. Fort, Jr., ed) Prentice-Hall,
Englewood Cliffs, N.J. 1962. Click here for
.pdf file.
- Survery article: C. McA. Gordon "Apsects of classical knot
theory", Knot Theory, Lecture Notes in Math, Vol 658,
Springer-Verlag, New York, 1978.
- Survery article: L.H. Kauffman "New invariants in the theory of
knots", Am. Math. Monthly 95 (1988), 195-242. Click here for
.pdf file.
- Survey article: W.B.R. Lickorish and K. Millett, "The new
polynomial
invariants of knots and links", Mathematics Magazine 61
(1988), 3-23.
- Useful reference on 3-manifolds and incompressibility: Allen Hatcher "Notes on Basic 3-Manifold Topology". Available as a .pdf file on his homepage.
- Useful reference on 2 and 3-dimensional geometries: Peter Scott "The Geometries of 3-manifolds" Bull. London Math. Soc, 15 (1983) p. 401-487.
- Follow up from class:
- Notes on tangles are here. They will help with proving Brunnian links are nontrivial and rational knots are prime.
- Tips for writing up your reading project:
- Please (la)tex your essay on your reading project. It should be 5-10 pages. (No longer please! This means you won't have room to write about everything you have understood. Working out what to write and what to leave out is as much part of the project as the reading and writing. Ask me if you need some help.)
- I've emailed the class the latex file for the final homework assignment along with the xfig files for the picture in it. This gives you a template for a document with a picture.
- Note the use of "overpic". This enables you to add text or mathematics to your figures. The size and relative location of this writing will not change if you change the scale of the picture.
- Note the use of \usepackage{pdfsync} (It is commented out in the final homework file.) This allows you to move between what you are typing and what you are seeing as you tex. If you press ctrl then click, the area will be highlighed in the other file. Try it on longer documents. It really helps with editing
- References: Makes sure you write all your references correctly. For papers, it is not enough to state the name of the journal. I need to see the name of the paper and authors, what year it appeared, volume, page number, etc, etc. Ask me or take a look at some papers if you do not know how to do this. (If you are more adventurous you could try using Bibtex.)
- The American Mathematical Society has some great help for writing mathematics. If you go to their author resource center http://www.ams.org/authors/ you'll find "Author Tools". Two particularly useful tools are the MR Serials Abbreviation List and the Tex Resources.
- I can help people learn how to use xfig. Please set up an appointment with me.
- Homework Assignments:
Start with your class notes and the text book.
Note that I've denoted some problems as class exercises. These
problems usually ask for
you to calculate or discover something about many knots (for
example all knots with crossing number of 7 or less). The best way
to get the most out of these questions is for everyone to do one or
two knots and pool the results. Gerardo will coordinate these
calculations and collate the results.
- Syllabus (last modified 3/5/07):
The references for each topic are given, where K&L refers to Knots and
Links by Peter Cromwell.
- Examples of knots and links, definition, knot equivalence. K&L Ch 1.
- Knot invariants, first examples. K&L Ch 1.9 and class notes.
- Combinatorial equivalence, knot diagrams, Reidemeister moves,
crossing number, linking number. K&L Ch 2.12 and Ch 3.
- Knot colorability. See handout above. Livingston "Knot
Theory" Chapter 3 sections 1 through 4. Gilbert and Porter "Knots
and Surfaces" Chapter 1.1.
- Surfaces, Euler characteristic and Genu. K&L Ch 2 sections 6 and 7
- Spanning surface , Seifert surface and genus of a knot. Genus-1 knots, surgery equivalence, genus and factorization K&L Ch 5 sections 1-6.
- Separation and Schoenflies Theorem, surgery and compression, transverse intersection and general position. K&L Ch 2 sections 4,5,8,9,10.
- Link constructions, satellite knots and companion knots, prime links and unique factorization. K&L Ch 4 sections 1-6.
- Hyperbolic knots, n-tangles, rational knots and 2-bridge knots. K&L Ch 4 sections 8-10.
- Overview of first homology group of a surface. K&L Ch 6 sections 1-4.
- Seifert matrix and matrix invariants; signature, determinant. K&L Ch 6 sections 5-7.
- Alexander-Conway polynomial. K&L Ch 7
- Rational tangles and continued fractions. K&L Ch 8.
- Kauffman bracket.
- Braids.
- Fundamental group of the knot complement.
- Introduction to geometric knot theory.
- Project Talks.
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