Math 150: Knot Theory Fall 2009 Tufts University
Professor
Dan Margalit, BP 209.
Text
Knot Theory, by Charles Livingston.
Office Hours
Tues. and Thurs., 1:30 to 3:00 pm.
Homework
The homework will be assigned weekly. In general, it will be handed out on Thursdays and due on Tuesdays. The problems will involve both computations and proofs.
Exams
Midterm Exam October 22 Final Exam December 10
Grading
The final grade for each student will be computed using whichever of the following schemes results in a higher number.
Scheme 1 Scheme 2 Homework 25% 25% Midterm Exam 35% 25% Final Exam 40% 50%
Syllabus
If you are handed a knot, how do you know if it is possible to untangle it? This is the overarching question in knot theory, which began in the 1880s. Consider, for instance, the knot shown on the other side of this page.
Knot Basics. We introduce the notion of knots and their diagrams. We will show that any two diagrams for the same knot differ by a finite sequence of “moves” called Reidemeister moves. As a consequence, we will be able to show that the most famous knot, the trefoil knot, is a nontrivial knot. This uses the notion of coloring.
Once we know that there exist nontrivial knots, we would like to come up with lots of examples of knots. Knot theorists have made tables of all knots with at most 16 crossings (there are 1,701,936 of them), and we will give several ways of listing knots, in particular, alternating knots, torus knots, satellite knots, pretzel knots, and closed braids and tangles. We will also introduce the Dowker and Conway notations for writing down knots.
Geometric, Algebraic, and Combinatorial Techniques. Now we really get down to the business of telling knots apart. The first important breakthrough was made in 1927 by Alexander; his discovery is now known as the Alexander polynomial. We can use this to show that the figure eight knot is nontrivial, that many pretzel knots are nontrivial, and that the (2,n)torus knots give an infinite family of distinct knots.
Next, we define the genus of a knot, and show that any knot can be broken up uniquely into a “sum” of prime knots, and that there are infinitely many primes.
Finally, we find a way of associating an abstract group to every knot. We do this not by focusing on the knot itself, but by the space around the knot. This technique can be used to show that almost all torus knots are distinct.
Numerical Knot Invariants. Given a diagram of a knot, it is easy to associate a number to it. However, this is only useful if, when you use a different diagram for the same knot, the number you come up with is the same. In this case, the number is called an invariant of the knot. One way to tell knots apart is to show that they have different invariants. The classical numerical invariants are: rank, determinant, genus, signature, braid index, bridge number, stick number, crossing number, and unknotting number. We will use signature, for example, to show that the lefthanded and righthanded trefoils are different knots.
Polynomial Knot Invariants. The first major advance in knot theory after the Alexander polynomial was the discovery of the Jones polynomial by Vaughn Jones in 1984. We will discuss the Kauffman bracket polynomial, the ConwayAlexander polynomial, the Jones polynomial, and the HOMFLY polynomial. None of these is a “complete” knot invariant (that is, you can find a nontrivial knot with trivial polynomial), however, these techniques were the main tool used in classifying the knots with up to 16 crossings. It is conjectured that the more modern Vassiliev invariants of knots (also known as “finite type invariants”) do give a complete invariant.