MATH 139

TOPICS IN KNOT THEORY: FINAL PROJECTS
  • Back to course webpage:

  • Overview: A chance to read in depth about a part of knot theory that interests you. (Suggested topics can be found below.) You will write an essay about your project. You will also give a 20 minute talk to the class about your topic. This should be at the level of the math table talks. The preparation grade will come from your meeting with me throughout the semester as you work on your project.
  • Grading policy: The final project is worth 50% of your course grade. The final project grade is made of three parts - preparation: 10%, talk: 15%, essay: 25%.

  • Preparation: The aim is for you to be reading/thinking/writing about your project throughout the semester. The final project is not meant to be a task that you complete at the last minute! In order to get full credit for preparation, you must meet with me (at least) at the following times:
    • The start of semester to discuss your interests.
    • During the last week of February to pick a topic.
    • Three times during March/April to discuss your progress, any questions you have and the content of your essay.
    • One additional meeting during April/May to discuss your talk.


  • The essay: The essay is a chance for you to write about what you have learned. It should be 5-10 pages long (and ideally written using latex). It must be submitted by the last day of class. You must also submit a half to one page outline of your essay by 5pm on Friday 23rd March (before Spring break). The outline will consist of a statement about the topic you are reading, the references you are using and a very brief outline of the structure of your essay.

  • The talk: The talks will be given during the last week of class or during reading period. It should be 20 minutes long and at the level of a math table talk. Winston Churchill once remarked "I'm sorry this is a long speech, I didn't have time to write a short one." The aim of the talk is for you to practice speaking about mathematics in front of a friendly audience. You will also practice the skill of saying less! In your talk, you should give an overview of your final project to your classmates explaining what you have discovered and why it is interesting. (So giving a well chosen example illustrating the main ideas is better than giving the technical details of a proof. I'll read these in your essay.)
SAMPLE TOPICS (last modified 2/1/07)

These are just a small sample of possible projects you might take on. In most cases, I've just given the initial reference for the project. Once you find something you are interested in, we'll add more material depending on your background and interests.
  • Wild Embeddings: Explore wild embeddings of the unit circle and sphere in three space. This includes Antoine's necklace and Antoine's horned sphere.
    • Prerequisite: Math 131
    • References: Rolfsen "Knots and Links" and R.H. Fox "A remarkable simple closed curve" Ann. of Math. Vol 50 (1949) pp 264, 265.
  • Creating Knot tables:
    • Prerequisite: open to everyone
    • References: Various articles by M. Thistlethwaite and others.
  • Colorability and the knot group: What does the fundamental group of a knot have to do with the knot's colorability? It turns out that the existence of a surjective map between the knot group and the dihedral group D_{n} determine whether the knot is n-colorable.
    • Prerequisite: Algebra at the level of Math 101 would do, plus familiarity of the knot group at a computational level.
    • References: Knots and Links by Livingston and Knots and Surfaces by Gilbet and Porter.
  • Colorability and torsion invariants: The determinant and mod p rank of a knot are captured by stronger invariants - torsion invariants.
    • Prerequisite: Algebra at the level of Math 122 (and possibly Math 123)
    • References: Livingston "Knots and Links" and Reidemeister "Knotentheorie"
  • Constructing lens spaces: A surgery along a knot in S^{3} is the result of "cutting off" a tubular neighborhood of the knot out of S^{3} and glueing the cut-out torus back in using some glueing homeomorphism. Surgery along a link is defined si milarly, doing this for each component of the link. This project aims to understand how lens spaces arise as the result of a surgery along a link, using a glueing homeomorphism determined by the continued fraction form of p/q.
    • Prerequisites: Math 131 and basic familiarity with the definition of a manifold. You don't need much knowledge of manifolds since everything happens in S^{3}
    • References: Knots, Links, Braids and 3-manifolds by V.V. Prasolov and A.B. Sossinsky.
  • Vassiliev Invariants: Learn what these invariants are and understand the connection between these and other knot polynomials.
    • Prequisites: open to everyone
    • References: Start with Knot Theory by Vassily Manturov and go from there.
  • Energy of Knots There are several projects to do here. One on knot energy modelling electric charge and one on knot energy modelling knots in ropes.
    • Prequisites: open to everyone
    • References: Start with Energy of Knots and Conformal Geometry by Jun O'Hara, other papers will be added as needed.
  • Knots and the topology of DNA:
    • Prequisites: open to everyone
    • References: Start with D.W. Sumner "Lifting the curtain: using topology to probe the hidden actin of enzymes", Notices AMS (42), 5, 1995. D.W. Sumner "Untangling DNA" Math Intell. 12(3) (1990) pp. 71-80.
  • Knots and total curvature: The total curvature of a knot is at least 4\pi.
    • Prequisites: open to everyone
    • References: J.W.Milnor "On the total curvature of knots." Ann. of Math (2) 52 (1950) pp. 248-257 and I. Fary "Sur la coubure totale d'une courbe gauche faisant un noeud." Bull. Soc. Math. France v. 77 (1949) pp. 128-138.
  • Geometry of Knots: There are more projects based on this theme. These are open to everyone. Please ask me for suggestions.
  • Knots and fundamental groups: Learn about the fundamental group of the complement of a knot. This is a very powerful invariant. Students who have not taken 131 will enjoy learning about the fundamental group of a knot and Van Kampen's Theorem. Students who have more experience will enjoy learning how to find the Alexander polynomial from the knot group presentation.
    • Prequisites: open to everyone
    • References: Start with Gilbert and Portrer "Knots and Surfaces", also Crowell and Fox "Introduction to Knot Theory".
  • Knots and Physics: Several topics to choose from, such as understanding how link diagrams can be interpreted as abstract tensor diagrams. Study the diagrammatic approach to the Yang-Baxter equation and its solutions.
    • Prequisites: open to everyone
    • References: Start with L. Kauffman "Knots and Physics".
  • Braids and the word problem: How to recognise braids, several projects are possible.
    • Prequisites: Math 122 and possibly 123.
    • References: Start with V. Manturov "Knot Theory" then move on to papers.
  • Braids and cryptography:
    • Prequisites: Math 122 and possibly 123.
    • References: Patrick Dehorney "Braid-based cryptography" Contemp. Math. 360 (2004) pp. 5-33. K.Marlburg "Overview of braid group cryptography".
  • Knots and Statistical mechanics: Explore the connections between knot invariants and statistical mechnical models, including vertex models, Potts-type models.
    • Prequisites: Open to everyone
    • References: V. Jones "Knot theory and Statistical Mechanics" Sci. American 263(5) (1990) pp. 98-103 and V. Jones "On knot invariants related to some statistical mechanical models." Pac. J. Math. 137(2) (1990) pp. 311-334
  • Dehn's Lemma and the Loop Theorem:
    • Prequisites: Math 272a
    • References: Rolfsen "Knots and Links" and J.Stallings "On the Loop Theorem" Ann. of Math. Vol 72, no.1 July 1960.
  • Obtaining 3-manifolds by surgery Every closed connected orientable 3-manifold can be obtained by "surgery" on S^3.
    • Prequisites: Math 272a (This project will require background reading.)
    • References: Start with R.Lickorish "An introduction to Knot Theory"
  • The knot quandle: Quandles are algebraic objects closely related to groups and algebras. It turns out that one can naturally associate a quandle to a knot. This gives a new invariant for knots.
    • Prequisites: open to everyone
    • References: Start with Manturov "Knot Theory" also D.Joyce "A classifying invariant of knots, the knot quandle", J. Pure Appl. Alg. v. 23, pp. 37-65.
  • The Jones polynomial and Khovanov homology: This project is the about one of the latest very interesting combinatorial development in knot theory.
    • Prequisites: Math 272a plus possibly some background reading.
    • References: Start with Manturov "Knot Theory". Then papers by Dror Bar-Natan.
  • Knots are determined by their complements: A central result in knot theory.
    • Prequisites: Math 272a plus background reading. This is a difficult project.
    • References: C.M.Gordon and R.A.Litherland "Knots are determined by their complements" J. Amer. Math. Soc. v. 2 (1989) pp. 371-415.