| Week |
Date of Monday |
Topics |
Example sheet |
Lecture notes |
Comments |
| 1 |
Jan. 5 |
Basic definitions. Isotopy. Isotopy invariants. |
One |
|
Make knots out of old shoelaces, sticks,
DNA,
Tangles,
or Klixx
(as seen in class!) |
| 2 |
Jan. 12 |
Coloring. Checkerboards. Martix of coloring equations
and the determinant. |
Two |
|
KnotInfo and
Knot
Atlas. |
| 3 |
Jan. 19 |
Determinants. The coloring group. Examples. |
Three |
|
The coloring group is the first homology of the branched
double cover [Introduction to Knot Theory, Crowell and Fox]. |
| 4 |
Jan. 26 |
Mirrors. Inversions. Codes. Alexander polynomials. |
Four |
|
Hoste, Thistlethwaite, and Weeks use DT codes to find the
first 1,701,936
knots. |
| 5 |
Feb. 2 |
Alexander polynomials and connect sums. Bridge position. Plats. |
Five |
|
Bridge presentation can be exponentially more complicated than
plat presentation. |
| 6 |
Feb. 9 |
Flypes. Braids (generators and relations) and their closures. Seifert circles. |
Six |
|
A braid applet by
Stephen Bigelow. |
| 7 |
Feb. 16 |
Every knot is isotopic to a braid closure.
Kauffman states. Kauffman polynomial. |
Seven |
|
Kauffman's webpage. Alexander's
paper on
the Alexander polynomial. |
| 8 |
Feb. 23 |
Jones polynomial. Span and crossing number. |
Eight |
|
An on-line calculator
for the Jones polynomial. |
| 9 |
Mar. 2 |
Tangles. Surfaces. |
Nine |
|
Slides
from a lecture by John Conway. (He uses a different sign
convention from Sanderson's.) Conway's ZIP proof, by
Francis and Weeks. |
| 10 |
Mar. 9 |
Knot genus and the Alexander polynomial. Conway, HOMFLY
polynomials. Relative strength of invariants. Video. |
Ten |
|
Stills from the Not
Knot video. Lots of knots,
collected by Bar-Natan. |
