Ukraine Canada Summer School 2006 Talk I

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Also see Some Non Obvious Examples.

Questions

  • Is the trefoil knot really knotted?
  • Is the trefoil knot equivalent to its mirror image?
  • Are K11n34 (the "Conway" knot) and K11n42 (the "Kinoshita-Terasaka" knot) really different?
K11n34.png K11n42.png
  • Which of these two is the knot at the gate of the Cambridge University maths department?
Gateknot-negated.jpg
  • Can you make a list of all knots?
  • The trefoil knot is bounds a Seifert surface. Do all knots do?
  • The complement of the trefoil knot is "fibered" with Seifert surfaces (see animation). Is that true for all knots? How does one decide?
  • Which knots are ribbon knots?
  • Which knots are slice knots?
  • And always, the hardest and most important question in mathematics: Why should we care??

Some Answers

  • 3-colourings.
  • The Kauffman bracket and the Jones polynomial.