||Date of Monday
||Knots, links and diagrams. Isotopy. Isotopy invariants.
is right-handed, as are most sugars.
You can make knots out of old shoelaces, sticks,
(as seen in class!)
||Coloring. Checkerboards. Matrix of coloring equations.
||Every connected 4-valent graph in the plane is the projection of
some link. KnotInfo and Knot
||Determinants. The coloring group. Examples.
||The coloring group is the first homology of the branched
double cover [Introduction to Knot Theory, Crowell and Fox].
||Mirrors. Inversions. Codes. Alexander polynomials.
||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
||Alexander polynomials and connect sums. Bridge position. Plats.
||Bridge presentation can be exponentially more complicated than
||Flypes. Standard position for 4-plats. Braids (generators
and relations) and their closures.
||A braid applet by
||Seifert circles. Every knot is isotopic to a braid closure.
Kauffman bracket. Kauffman states.
||Kauffman's webpage. Alexander's
the Alexander polynomial.
||Kauffman polynomial. Jones polynomial. Span and crossing
||An on-line calculator
for the Jones polynomial.
||Tangles. Surfaces. Knot genus.
from a lecture by John Conway. (He uses a different sign
convention from Sanderson.) Conway's ZIP proof, by
Francis and Weeks.
||Additivity of knot genus under connect sum. Torus knots.
Conway, HOMFLY polynomials. Relative strength of
||Stills from the Not
Knot video. Lots of knots,
collected by Bar-Natan.