Knot Theory, Knot Practice

Freshman Advising Seminar 18.A39

Wednesdays 3-5pm
Room 2-136

Advisor: Kyle Ormsby (, office 2-275)
Associate Advisor: Danny Shi (

Textbook: The Knot Book by Colin C. Adams, available through the AMS, online booksellers, and the Coop. (The cheapest I've found so far was on a South American river....)

Syllabus [pdf]

The earbuds in your bookbag know all about them -- do you? Knots pervade culture, science, and technology. They also inhabit a beautiful niche in mathematics that we'll examine in this seminar.

Tie a figure-eight in a rope and fuse the ends. Can you untie the twisted loop into a circle? If you think it's impossible but your friend thinks there's hope, how could you convince her? One persuasive proof (in all of math, but especially in knot theory) might involve assigning a number (or generlization of a number!) to your knot that doesn't change as you attempt to untie it. Such a quantity is called an invariant. If the invariant of the figure-eight differs from the unknot (circle), then you can convince your friend!

We will construct and compute knot invariants in this seminar. There are no prerequisites beyond an interest in knots.


  • Shi An
  • Ravi Bajaj
  • Yongquan Lu
  • Scott McDonald
  • Marissa Stephens
  • Anderson Wang
  • Patrick Yang
  • Angel Yu


The regular class meetings will occur on Wednesdays from 3:00 to 5:00pm in 2-136. Students will lead discussions of relevant mathematical topics taken from Colin Adams's The Knot Book and other sources.

Sep 5 - First meeting: Perspectives on knots, Reidemeister moves
Sep 12 - Scott and Anderson: Knots, links, and orientations
Sep 19 - Ravi and Marissa: Rational tangles
Sep 26 - Patrick and Angel: Unknotting, bridge, and crossing numbers
Oct 3 - Shi and YQ: Surfaces, Euler characterstic, and genus
Oct 5 - Picnic at 3pm in Killian! (Pictures)
Oct 10 - YQ and Anderson: Additivity of the genus, torus knots
Oct 17 - Ravi and Angel: Hyperbolic knots and knot complements
Oct 24 - Marissa and Patrick: Braids
Oct 31 - Shi and Scott: Bracket and Jones polynomials
Nov 7 - Angel and Ravi: The Jones polynomial and alternating links
Nov 14 - Marissa and Patrick: Knots, graphs, and the Arf invariant
Nov 21 - Happy Thanksgiving!
Nov 28 - Shi and YQ: Total curvature (Milnor's article)
Dec 5 - Anderson and Scott: Racks and quandles (Nelson's article)
Dec 12 - Group activity (TBA)

Kyle M. Ormsby
Department of Mathematics 2-275
Massachusetts Institute of Technology
Cambridge, MA 02139