| Week |
Date of Monday |
Topics |
Example sheet |
Comments |
| 1 |
Jan. 11 |
Knots, links and diagrams. Isotopy. Isotopy invariants. |
One |
DNA
is right-handed, as are most sugars.
You can make knots out of old shoelaces, sticks,
DNA,
Tangles,
or Klixx
(as seen in class!) |
| 2 |
Jan. 18 |
Coloring. Checkerboards. Matrix of coloring equations. |
Two |
Every connected 4-valent graph in the plane is the projection of
some link. KnotInfo and Knot
Atlas. |
| 3 |
Jan. 25 |
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 |
Feb. 1 |
Mirrors. Inversions. Codes. Alexander polynomials. |
Four |
How to tie your shoelaces,
including a discussion
of the reef and granny knots.
Hoste, Thistlethwaite, and Weeks use DT codes to find the first
1,701,936
knots. |
| 5 |
Feb. 8 |
Alexander polynomials and connect sums. Bridge position. Plats. |
Five |
Bridge presentation can be exponentially more complicated than
plat presentation. |
| 6 |
Feb. 15 |
Flypes. Standard position for 4-plats. Braids (generators
and relations) and their closures. |
Six |
A braid applet by
Stephen Bigelow. |
| 7 |
Feb. 22 |
Seifert circles. Every knot is isotopic to a braid closure.
Kauffman bracket. Kauffman states. |
Seven |
Kauffman's webpage. Alexander's
paper on
the Alexander polynomial. |
| 8 |
Mar. 1 |
Kauffman polynomial. Jones polynomial. Span and crossing
number. |
Eight |
An on-line calculator
for the Jones polynomial. |
| 9 |
Mar. 8 |
Tangles. Surfaces. Knot genus. |
Nine |
Slides
from a lecture by John Conway. (He uses a different sign
convention from Sanderson.) Conway's ZIP proof, by
Francis and Weeks. |
| 10 |
Mar. 15 |
Additivity of knot genus under connect sum. Torus knots.
Conway, HOMFLY polynomials. Relative strength of
invariants. Video. |
Ten |
Stills from the Not
Knot video. Lots of knots,
collected by Bar-Natan. |
