Knot Theory, Knot Practice

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!

Participants

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

Schedule

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)