Knot Theory

UNH, Spring 2010

Mondays, 11:40-12:30

Schedule: Here are the dates, topics and speakers for the rest of the semester

Feb 1: Reidemeister moves, 3-colorability and the Jones polynomial. Emily
Feb 8: The HOMFLYPT, Kauffman and maybe Alexander polynomials. Emily
Feb 15: Khovanov homology: an overview. Emily
March 1: Khovanov homology: definitions. Tim. Notes
March 8: Khovanov homology: homology, what is it? Ian Rebecca's Notes
March 15: Spring break!
March 22: Khovanov homology: proof of invariance. Rebecca. Notes
March 29: Khovanov homology and TQFTs. Dmitri. Justin's notes, Dmitri's notes
April 5: Thurston: overview and surfaces/Euler characteristic. Emily. Chad's Notes
April 12: Thurston: manifolds and hyperbolic space. Chad. Emily's Notes
April 19: Thurston: gluing tetrahedra to construct the complement of the figure-8 knot. William. William's Notes
April 26: Thurston: compliments of other knots. Brianna. Emily's Notes
Knots and computers. Geoff
May 10: Statement of volume conjecture, and "Not Knot." Emily and the Geometry Center respectively.

Course Summary: If you've ever played with one of those puzzles where you have to remove a ball from a rope wound around a wooden base, a brass ring, and a wine bottle, then you understand that the question of whether two knots are the same is frustratingly difficult and fascinating.

A big tool in knot theory is knot invariants: maps from the set of knots to a more reasonably behaved set. We'll begin with the Jones polynomial, which is a polynomial created from a knot in a reasonably straightforward way. The first two thirds of this course will be about the Jones polynomial and its generalizations: the Kauffman and HOMFLYPT polynomials, and a cool new gadget called Khovanov homology. The final third of this course will have a slightly different focus; we'll study the more geometric problem of understanding a knot's compliment in three-space. We won't do this in much generality, but will be content to understand one or two examples well.

Along the way, we'll discuss some fascinating open problems, starting with "do any non-trivial knots have trivial Jones polynomial?" and ending with the volume conjecture (relating the volume of the complement of a hyperbolic knot to its Jones polynomial).

Bibliography: Our introduction to knots and polynomial invariants will be largely based on Colin Adam's "The knot book," and Lickorish's "An Introduction to Knot Theory." For Khovanov homology, we will turn to Dror Bar-Natan's "On Khovanov's categorification of the Jones polynomial." This is one of the best-written math research paper I know of, and I can't resist quoting the entirety of its abstract: "The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of "the categorification of the Jones polynomial". For the same low cost we also provide some computations, including one that shows that Khovanov's invariant is strictly stronger than the Jones polynomial and including a table of the values of Khovanov's invariant for all prime knots with up to 11 crossings."

Finally, we turn to Bill Thurston's "Three-dimensional geometry and topology" to calculate the geometry of the complement of the figure-8 knot. Time permitting, we'll also discuss Thistlethwaite's "Links with trivial Jones polynomial."

I have copies of all of these books and papers, which you are encouraged to borrow instead of purchasing your own.