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P,q Trefoil Knot

  1. Dec 6, 2012 #1
    So in the process of giving us a crude definition of a trefoil knot, our professor talks a bit about a function on a torus.

    If we view the torus as the identification of sides of a square, and define a function y=(p/q)x, then we may only go from the bottom left corner (0,0) to the top right corner (1,1) (I guess forming a knot) if (p,q)=1. Two questions...

    1) Isn't any function y=(m/n)x the same as a function y=(p/q)x with (p,q)=1?

    2) Is there a simple way to understand why the (p,q) must be 1, or is it something not so trivial?
     
  2. jcsd
  3. Dec 6, 2012 #2

    jedishrfu

    Staff: Mentor

  4. Dec 6, 2012 #3
    I should mention, my algebra and algebraic topology isn't great. I've studied a good bit of a semester of introductory undergraduate group theory, but not much else. The treatment of knots in my topology class is purely as an example of Van Kampen's theorem and not as a topic in and of itself, so I know very little about them.
     
  5. Dec 7, 2012 #4

    lavinia

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    yes.
    it's just that if p and q are relatively prime then they tell you the actual number of rotations around the torus. p/q is just the slope of a straight line on the flat torus. Any common factors cancel out.
     
  6. Dec 7, 2012 #5
    So if we take p and q not relatively prime rotations around the torus, what happens?
     
  7. Dec 9, 2012 #6

    lavinia

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    try the case of a (2,2) curve. You can draw a picture on a rectangle with opposite edges identified.
     
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