MA3F2, Knot Theory
Winter 2010


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.