MA3F2, Knot Theory
Winter 2009


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.