Knots and 3-manifolds - Summer Tutorial 2002

• Instructor: Ciprian Manolescu
• Time and Place: TuTh 8-9:30 pm in Science Center 103b
• Homepage: http://www.people.fas.harvard.edu/~manolesc/summer.html
• E-mail: manolesc_at_fas.harvard.edu
• Office: Science Center 431f
• Office Hours: Tuesdays and Thursdays 10:30 - 11:30 am and by appointment

Homework:

• Problem set 1 in pdf.
• Problem set 2 in pdf.
• Problem set 3 in pdf.
• Problem set 4 in pdf.
• Problem set 5 in pdf.

Outline:

This tutorial will introduce the theory of knots and 3-dimensional manifolds. We will start by addressing the question: How can one prove that two knots can or cannot be deformed into each other? We will talk about several knot invariants, such as the Alexander and the Jones polynomials. Then, we will move on to discuss four different procedures for constructing 3-dimensional manifolds: Heegard splittings, surgery, branched coverings and geometric decompositions. The first three of these are related to knot theory, while the fourth makes use of differential geometry. We will also study Seifert fibrations and enumerate the eight 3-dimensional geometries. One goal is to understand the importance of Thurston's geometrization conjecture for the classification of 3-manifolds.

Requirements:

By late July each student should choose a particular topic on which to write a short paper (5-7 pages) and give an in-class presentation. The papers are due on the last day of classes (probably August 15). They can fulfill your junior paper requirement. You are also welcome (and encouraged) to submit a draft before your presentation if you'd like.

Organization:

In the first four weeks of the course, there will be lectures and optional homeworks. Near the end the classes will consist of student presentations (45 minutes each).

Topics to be covered in the first part of the course:

• plane diagrams of knots and links, Reidemeister moves;
• Seifert surfaces;
• knot invariants: the fundamental group, the Alexander and the Jones polynomials;
• Heegard splittings of 3-manifolds;
• surgery on links and Kirby calculus;
• 3-manifolds as branched coverings;
• prime decompositions;
• Seifert fibrations;
• geometric structures on 3-manifolds and a discussion of Thurston's geometrization conjecture.

If we run out of time, some of these may turn into possible student project topics.

Additional suggestions for project topics:

• the HOMFLY polynomial;
• the Jones polynomial and the geometry of alternating links;
• Dehn's lemma, the loop and the sphere theorems;
• 2-dimensional orbifolds and Seifert fibrations;
• Reidemeister torsion used to distinguish between homotopy equivalent lens spaces;
• constructions of hyperbolic 3-manifolds;
• surface bundles and pseudo-Anosov diffeomorphisms;
• incompressible surfaces, hierarchies, and Waldhausen's theorem;
• contact structures on 3-manifolds.

References:

• D. Rolfsen, "Knots and links." Publish or Perish, 1990.
• N. Saveliev, "Lectures on the topology of 3-manifolds." Berlin, New York: de Gruyter, 1999.
• A. Kawauchi, "A survey of knot theory." Basel: Birkhauser Verlag, 1996.
• W.B.R. Lickorish, "An introduction to knot theory." New York: Springer, 1997.
• W. Thurston, "Three-dimensional geometry and topology," edited by Silvio Levy. Princeton University Press, 1997.
• J. Hempel, "3-manifolds." Princeton: Princeton Univ. Press, 1976.
• P. Scott, "The geometries of 3-manifolds." Bull. London Math. Soc. 15 (1983), no. 5, 401--487.
• F. Bonahon, Geometric Structures on 3-manifolds, available online at http://math.usc.edu/~fbonahon/Research/Preprints/Preprints.html
• A. Hatcher, Notes on Basic 3-Manifold Topology, available online at http://www.math.cornell.edu/~hatcher/#3M

Please don't recall a book from Cabot if you see it is checked out. This means that I have it !!! I would be glad to lend it to you for consultation. Additional copies can also be found in Birkhoff library.

A knot theory joke from "The Knot Book" by Colin Adams:

A woman walks into a bar accompanied by a dog and a cow.

The bartender says, "Hey, no animals are allowed in here."

The woman replies, "These are very special animals."

"How so?"

"They're knot theorists."

The bartender raises his eyebrows and says, "I've met a number of knot theorists who I thought were animals, but never an animal that was a knot theorist."

"Well, I'll prove it to you. Ask them them anything you like."

So the bartender asks the dog, "Name a knot invariant."

"Arf, arf" barks the dog.

The bartender scowls and turns to the cow asking, "Name a topological invariant."

The cow says, "Mu, mu."

At this point the bartender turns to the woman, says, "Just what are you trying to pull" and throws them out of the bar.

Outside, the dog turns to the woman and asks, "Do you think I should have said the Jones polynomial?"

Remark:

This is why in this course we will spend some time discusing the Jones polynomial and we might mention the Arf and mu invariants only briefly!